INTERMITTENT DEMAND FORECASTING WITH INTEGER AUTOREGRESSIVE MOVING AVERAGE MODELS A Thesis submitted for the degree of Doctor of Philosophy By Maryam Mohammadipour School of Business and Management Faculty of Enterprise and Innovation Buckinghamshire New University Brunel University April, 2009 ii Abstract This PhD thesis focuses on using time series models for counts in modelling and forecasting a special type of count series called intermittent series. An intermittent series is a series of non-negative integer values with some zero values. Such series occur in many areas including inventory control of spare parts. Various methods have been developed for intermittent demand forecasting with Croston?s method being the most widely used. Some studies focus on finding a model underlying Croston?s method. With none of these studies being successful in demonstrating an underlying model for which Croston?s method is optimal, the focus should now shift towards stationary models for intermittent demand forecasting. This thesis explores the application of a class of models for count data called the Integer Autoregressive Moving Average (INARMA) models. INARMA models have had applications in different areas such as medical science and economics, but this is the first attempt to use such a model-based method to forecast intermittent demand. In this PhD research, we first fill some gaps in the INARMA literature by finding the unconditional variance and the autocorrelation function of the general INARMA(p,q) model. The conditional expected value of the aggregated process over lead time is also obtained to be used as a lead time forecast. The accuracy of h-step-ahead and lead time INARMA forecasts are then compared to those obtained by benchmark methods of Croston, Syntetos-Boylan Approximation (SBA) and Shale-Boylan- Johnston (SBJ). The results of the simulation suggest that in the presence of a high autocorrelation in data, INARMA yields much more accurate one-step ahead forecasts than benchmark methods. The degree of improvement increases for longer data histories. It has been shown that instead of identification of the autoregressive and moving average order of the INARMA model, the most general model among the possible models can be used for forecasting. This is especially useful for short history and high autocorrelation in data. The findings of the thesis have been tested on two real data sets: (i) Royal Air Force (RAF) demand history of 16,000 SKUs and (ii) 3,000 series of intermittent demand from the automotive industry. The results show that for sparse data with long history, there is a substantial improvement in using INARMA over the benchmarks in terms of Mean Square Error (MSE) and Mean Absolute Scaled Error (MASE) for the one- step ahead forecasts. However, for series with short history the improvement is narrower. The improvement is greater for h-step ahead forecasts. The results also confirm the superiority of INARMA over the benchmark methods for lead time forecasts. iii Acknowledgements Completing a PhD is by far the most challenging yet rewarding task I have ever undertaken. Fulfilling this task was not possible without the support of numerous people, to whom I am grateful. Most importantly, I would like to express my deep and sincere gratitude to my supervisor Prof. John Boylan, Head of Research, Faculty of Enterprise and Innovation, Buckinghamshire New University. His in-depth knowledge, constant encouragement, support and particularly his advice have been invaluable in ensuring I was on the right track throughout the PhD studies. I appreciate all his contributions of time and ideas. I also wish to express my appreciation to Dr Aris Syntetos of Salford Business School, University of Salford, for providing me with the empirical data for this thesis. I would also like to thank Ms Laura Bray, Faculty Research Officer, Buckinghamshire New University for her sympathetic help she has provided during the last three years. I am grateful to the Buckinghamshire New University for the scholarship they provided for this PhD study and also the support to enable me to attend and present my academic papers at international conferences. My special appreciation goes to my parents, Mina Moradkhan and Gholamhossein Mohammadipour, who always encouraged me to concentrate on my studies. Their non-stop support has made my dreams come true. I also want to thank my brother, Dr Mani Mohammadipour, and my uncle, Dr Shahryar Mohammadipour, for their love and friendship. Last but not least, words cannot express my special thanks to my beloved husband, Soroosh Saghiri, for his endless love, faith and encouragement in fulfilling this aspiration. iv To Soroosh my husband my best friend v Contents Chapter 1 Introduction ............................................................................................ 1 1.1 Introduction ....................................................................................................... 2 1.2 Research Overview ........................................................................................... 3 1.3 Business Context ............................................................................................... 3 1.4 Research Background ....................................................................................... 4 1.5 Research Problem ............................................................................................. 6 1.5.1 Initial Problem .............................................................................................. 6 1.5.2 Research Questions ....................................................................................... 6 1.6 Research Methodology ..................................................................................... 8 1.6.1 Research Philosophy ..................................................................................... 9 1.6.2 Research Approach ....................................................................................... 9 1.6.3 Research Strategy ......................................................................................... 9 1.6.3.1 Mathematical Analysis ......................................................................... 9 1.6.3.2 Simulation .......................................................................................... 10 1.6.3.3 Empirical Analysis ............................................................................. 11 1.7 Thesis Structure .............................................................................................. 11 Chapter 2 Forecasting Intermittent Demand ....................................................... 13 2.1 Introduction ..................................................................................................... 13 2.2 Definition of Intermittent Demand ................................................................. 14 2.3 Methods of Forecasting Intermittent Demand ................................................ 17 2.3.1 Croston?s Method ....................................................................................... 18 2.3.1.1 Bias Correction for Bernoulli Demand Incidence .............................. 20 2.3.1.2 Bias Correction for Poisson Demand Incidence ................................. 21 2.3.1.3 Modified Croston (MC) ...................................................................... 22 2.3.1.4 Models underlying Croston?s Method ................................................ 23 2.3.2 Bootstrapping .............................................................................................. 25 2.3.2.1 Snyder (2002) ..................................................................................... 25 2.3.2.2 Willemain et al. (2004) ....................................................................... 26 2.3.2.3 Porras and Dekker (2007) ................................................................... 27 2.3.3 Causal Models ............................................................................................ 28 2.3.4 Conclusions on IDF Methods ..................................................................... 30 2.4 Assessing Forecast Accuracy .......................................................................... 30 2.4.1 Absolute Accuracy Measures ..................................................................... 32 vi 2.4.2 Accuracy Measures Relative to another Method ........................................ 32 2.4.3 Conclusions on Accuracy Measures ........................................................... 34 2.5 Comparative Studies ....................................................................................... 36 2.6 INARMA Models ........................................................................................... 37 2.7 Conclusions ..................................................................................................... 38 Chapter 3 Integer Autoregressive Moving Average Models............................... 41 3.1 Introduction ..................................................................................................... 41 3.2 DARMA Models ............................................................................................. 43 3.2.1 DAR(1) Model ............................................................................................ 44 3.2.2 Applications of DARMA Models ............................................................... 45 3.3 INARMA Models ........................................................................................... 46 3.3.1 INAR(1) Model .......................................................................................... 47 3.3.2 INAR(2) Model .......................................................................................... 50 3.3.3 INAR(p) Model .......................................................................................... 52 3.3.4 INMA(1) Model .......................................................................................... 54 3.3.5 INMA(2) Model .......................................................................................... 56 3.3.6 INMA(q) Model .......................................................................................... 57 3.3.7 INARMA(1,1) Model ................................................................................. 58 3.3.8 INARMA(p,q) Model ................................................................................. 60 3.3.8.1 First and Second Unconditional Moments ......................................... 61 3.3.8.2 Autocorrelation Function (ACF) ........................................................ 62 3.3.9 Applications of INARMA Models ............................................................. 67 3.3.10 Aggregation in INARMA Models .......................................................... 70 3.3.10.1 Overlapping Temporal Aggregation .............................................. 72 3.3.10.2 Non-overlapping Temporal Aggregation ....................................... 73 3.3.10.3 Cross-sectional Aggregation .......................................................... 76 3.3.10.4 Over a Forecast Horizon Aggregation ........................................... 80 3.4 Summary of Literature Review ....................................................................... 81 3.5 Conclusions ..................................................................................................... 85 Chapter 4 Identification in INARMA Models ..................................................... 86 4.1 Introduction ..................................................................................................... 86 4.2 Testing Serial Dependence ............................................................................. 87 4.2.1 Runs Test .................................................................................................... 87 4.2.2 The Score Test ............................................................................................ 88 4.2.3 Portmanteau-type Tests .............................................................................. 88 4.3 Identification based on ACF and PACF ......................................................... 90 vii 4.3.1 Autocorrelation Function (ACF) ................................................................ 90 4.3.2 Partial Correlation Function (PACF) .......................................................... 91 4.3.2.1 PACF of an INAR(p) Model .............................................................. 91 4.3.2.2 PACF of an INMA(q) Model ............................................................. 94 4.3.2.3 PACF of an INARMA(p,q) Model ..................................................... 95 4.4 Residual Analysis ............................................................................................ 96 4.5 Identification based on Penalty Functions ...................................................... 99 4.6 The Identification Procedure ......................................................................... 101 4.7 Conclusions ................................................................................................... 102 Chapter 5 Estimation in INARMA Models ........................................................ 104 5.1 Introduction ................................................................................................... 104 5.2 Estimation in an INARMA(0,0) Model ........................................................ 106 5.3 Estimation in an INAR(1) Model.................................................................. 107 5.3.1 YW for INAR(1) ....................................................................................... 107 5.3.2 CLS for INAR(1) ...................................................................................... 108 5.3.3 CML for INAR(1) ..................................................................................... 109 5.3.4 Conditional GMM for INAR(1) ............................................................... 111 5.4 Estimation in an INAR(p) Model.................................................................. 112 5.4.1 YW for INAR(p) ....................................................................................... 112 5.4.2 CLS for INAR(p) ...................................................................................... 113 5.4.3 CML for INAR(p) ..................................................................................... 113 5.5 Estimation in an INMA(1) Model ................................................................. 115 5.5.1 YW for INMA(1) ...................................................................................... 115 5.5.2 CLS for INMA(1) ..................................................................................... 115 5.6 Estimation in an INMA(q) Model ................................................................. 116 5.6.1 YW for INMA(q) ...................................................................................... 116 5.6.2 CLS for INMA(q) ..................................................................................... 117 5.6.3 GMM based on Probability Generation Functions for INMA(q) ............. 117 5.7 Estimation in an INARMA(1,1) Model ........................................................ 119 5.7.1 YW for INARMA(1,1) ............................................................................. 119 5.7.2 CLS for INARMA(1,1) ............................................................................ 120 5.8 YW Estimators of an INARMA(2,2) Model ................................................ 121 5.9 Conclusions ................................................................................................... 122 Chapter 6 Forecasting in INARMA Models ....................................................... 124 6.1 Introduction ................................................................................................... 124 6.2 MMSE Forecasts ........................................................................................... 125 viii 6.2.1 MMSE Forecasts for an INAR(p) Model ................................................. 125 6.2.2 MMSE Forecasts for an INMA(q) Model ................................................ 127 6.2.3 MMSE Forecasts for an INARMA(p,q) Model ........................................ 128 6.3 Forecasting over Lead Time ......................................................................... 129 6.3.1 Lead Time Forecasting for an INAR(1) Model ........................................ 129 6.3.2 Lead Time Forecasting for an INMA(1) Model ....................................... 132 6.3.3 Lead Time Forecasting for an INARMA(1,1) Model ............................... 133 6.3.4 Lead Time Forecasting for an INARMA(p,q) Model ............................... 134 6.4 Conclusions ................................................................................................... 136 Chapter 7 Simulation Design ............................................................................... 137 7.1 Introduction ................................................................................................... 137 7.2 Rationale for Simulation ............................................................................... 138 7.3 Simulation Design ......................................................................................... 139 7.3.1 The Range of Series .................................................................................. 139 7.3.2 Producing INARMA(p,q) Series .............................................................. 140 7.3.3 Control Parameters ................................................................................... 141 7.3.3.1 INARMA Parameters ....................................................................... 141 7.3.3.2 Length of Series ................................................................................ 143 7.3.3.3 Forecast Horizon and Lead Time ..................................................... 143 7.3.3.4 Benchmark Methods? Parameters..................................................... 144 7.3.4 Identification Procedure ............................................................................ 144 7.3.5 Estimation of Parameters .......................................................................... 145 7.3.6 Forecasting Method .................................................................................. 146 7.3.7 Performance Metrics ................................................................................. 147 7.4 Verification ................................................................................................... 147 7.5 Conclusions ................................................................................................... 148 Chapter 8 Simulation Results .............................................................................. 149 8.1 Introduction ................................................................................................... 149 8.2 Details of Simulation .................................................................................... 150 8.3 Accuracy of INARMA Parameter Estimates ................................................ 151 8.4 Forecasting Accuracy of INARMA Estimation Methods ............................. 154 8.5 Croston-SBA Categorization ........................................................................ 161 8.6 INARMA vs Benchmark Methods ............................................................... 166 8.6.1 INARMA with Known Order ................................................................... 166 8.6.2 INARMA with Unknown Order ............................................................... 173 8.6.2.1 Identification among Two Processes ................................................ 174 ix 8.6.2.2 All-INAR(1) ..................................................................................... 178 8.6.2.3 Identification among Four Processes................................................ 180 8.6.2.4 All-INARMA(1,1) ............................................................................ 188 8.6.2.5 All-INAR(1) vs Benchmark Methods .............................................. 191 8.6.3 Lead Time Forecasts ................................................................................. 193 8.7 Conclusions ................................................................................................... 199 Chapter 9 Empirical Analysis .............................................................................. 202 9.1 Introduction ................................................................................................... 202 9.2 Rationale for Empirical Analysis .................................................................. 203 9.3 Demand Data Series ...................................................................................... 204 9.4 Design of Empirical Analysis ....................................................................... 206 9.5 INARMA vs Benchmark Methods ............................................................... 207 9.5.1 All-INAR(1) ............................................................................................. 208 9.5.2 All-INARMA(1,1) .................................................................................... 211 9.5.3 Identification among four Processes ......................................................... 214 9.5.4 INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) Series .............. 217 9.5.5 h-step-ahead Forecasts for INAR(1) Series .............................................. 222 9.5.6 The Effect of Length of History ............................................................... 226 9.5.7 Lead Time Forecasting for INAR(1) Series ............................................. 228 9.5.8 Lead Time Forecasting for all-INAR(1) ................................................... 234 9.6 Conclusions ................................................................................................... 238 Chapter 10 Conclusions and Further Research ................................................... 241 10.1 Introduction ................................................................................................... 241 10.2 Contributions ................................................................................................. 242 10.3 Conclusions from the Theoretical Part of the Thesis .................................... 243 10.3.1 The Unconditional Variance of an INARMA(p,q) Model ................... 244 10.3.2 The Autocorrelation Function of an INARMA(p,q) Model ................. 245 10.3.3 The YW Estimators of an INARMA(1,1) Model ................................. 246 10.3.4 Lead Time Forecasting of an INARMA(p,q) Model ............................ 246 10.4 Conclusions from the Simulation Part of the Thesis..................................... 247 10.4.1 The Performance of Different Estimation Methods ............................. 248 10.4.2 The Croston-SBA Categorization ......................................................... 248 10.4.3 Identification in INARMA Models ...................................................... 249 10.4.4 Comparing INARMA with the Benchmark Methods ........................... 250 10.5 Conclusions from the Empirical Part of the Thesis ...................................... 252 10.5.1 Identification in INARMA Models ...................................................... 252 x 10.5.2 The Performance of Different Estimation Methods ............................. 252 10.5.3 Comparing INARMA with the Benchmark Methods ........................... 253 10.5.4 The Problem with MASE ..................................................................... 253 10.6 Practical and Software implications .............................................................. 254 10.7 Limitations and Further Research ................................................................. 256 References..................................................................................................................... 260 Appendix 3.A Autocorrelation Function of an INARMA(1,1) Model ......................... 272 Appendix 3.B The Unconditional Variance of an INARMA(p,q) Model .................... 274 Appendix 3.C The Cross-Covariance Function between Y and Z for an INARMA(p,q) Model ............................................................................................................................. 280 Appendix 3.D Over Lead Time Aggregation of an INAR(1) Model ........................... 281 Appendix 4.A Infinite Autoregressive Representation of an INARMA(p,q) Model ... 286 Appendix 5.A The CLS Estimators of an INARMA(1,1) Model ................................. 289 Appendix 5.B The Unconditional Variance of an INARMA(2,2) Model .................... 291 Appendix 6.A Lead Time Forecasting for an INAR(2) Model .................................... 293 Appendix 6.B Lead Time Forecasting for an INARMA(1,2) Model ........................... 297 Appendix 6.C Lead Time Forecasting for an INARMA(p,q) Model ........................... 300 Appendix 8.A The MSE of YW and CLS Estimates for INAR(1), INMA(1) and INARMA(1,1) Processes ............................................................................................... 303 Appendix 8.B Impact of YW and CLS Estimates on Accuracy of Forecasts using MASE ....................................................................................................................................... 306 Appendix 8.C Croston-SBA Categorization for INAR(1), INMA(1) and INARMA(1,1) ....................................................................................................................................... 308 Appendix 8.D Comparing the Accuracy of INARMA Forecasts for all points in time and issue points .................................................................................................................... 311 Appendix 8.E Comparison of MASE of INARMA (known order) with Benchmarks 314 Appendix 8.F Comparison of six-step ahead MSE of INARMA (known order) with Benchmarks ................................................................................................................... 317 Appendix 8.G Comparison of all-INAR(1) and all-INARMA(1,1) ............................. 320 Appendix 8.H Comparison of MASE of INARMA (unknown order) with Benchmarks ....................................................................................................................................... 323 Appendix 8.I Comparison of MASE of INARMA with Benchmarks for Lead Time Forecasts ........................................................................................................................ 325 Appendix 9.A INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) Series of 16,000 Series ............................................................................................................................. 330 Appendix 9.B h-step-ahead Forecasts for the INARMA(0,0), INMA(1), and INARMA(1,1) Series..................................................................................................... 334 Appendix 9.C Lead Time Forecasts for the INARMA(0,0), INMA(1), and INARMA(1,1) Series..................................................................................................... 341 xi List of Illustrations Figure 1-1 The structure of Chapter 1 ............................................................................... 2 Figure 1-2 The structure of the research methodology ...................................................... 8 Figure 1-3 The structure of the thesis .............................................................................. 12 Figure 2-1 The categorization scheme for intermittent demand data (Syntetos et al. 2005) ......................................................................................................................................... 16 Figure 3-1 The covariance at lag , , when qkpk ?? , ........................................... 63 Figure 3-2 The covariance at lag at lag , , when qk pk ?? , ................................. 64 Figure 3-3 The covariance at lag , , when qk pk ?? , ........................................... 65 Figure 3-4 The covariance at lag , , when qkpk ?? , ........................................... 65 Figure 3-5 Non-overlapping temporal aggregation for ....................................... 74 Figure 3-6 The correlation between the INMA parts of the non-overlapping temporal aggregation of INAR(1) process over two periods .......................................................... 75 Figure 4-1 Time series plot of one demand series among 16,000 series ......................... 98 Figure 4-2 Correlograms of the selected series among 16,000 series ............................. 98 Figure 4-3 Correlograms of the residuals of the INAR(1) model ................................... 99 Figure 6-1 h-step-ahead forecast for an INAR(p) model when ph ? ........................... 126 Figure 6-2 h-step-ahead forecast for an INMA(q) model when .......................... 127 Figure 8-1 Cut-off values for Croston and SBA when (Syntetos et al., 2005) .. 162 Figure 3.B-1 in an INARMA(p,q) process when .............................. 276 Figure 3.B-2 in an INARMA(p,q) process when .............................. 277 Figure 3.B-3 in an INARMA(p,q) process when .............................. 278 k k? k k? k k? k k? 12?k qh ? 20.?? ),cov( ji ZY qp ? ),cov( ji ZY qp ? ),cov( ji ZY qp ? xii List of Tables Table 2-1 The categorization scheme for intermittent demand data (Eaves and Kingsman, 2004) ............................................................................................................. 15 Table 2-2 Example data (Hyndman, 2006)...................................................................... 34 Table 2-3 Comparing ES with ZF based on different accuracy measures ...................... 34 Table 2-4 Accuracy measures for simulation and empirical studies ............................... 35 Table 2-5 The categorization of methods of intermittent demand forecasting based on their assumptions ............................................................................................................. 38 Table 3-1 Review of count models in time series............................................................ 43 Table 3-2 Literature survey on Integer Autoregressive Moving Average models .......... 82 Table 5-1 Research papers on estimation of parameters of INARMA models ............. 105 Table 5-2 The relationship between the order of the model and the type of YW equation ....................................................................................................................................... 116 Table 6-1 Coefficients of in each of for an INAR(1) model ...................... 130 Table 6-2 Coefficients of in each of for an INAR(1) model ................. 131 Table 6-3 Coefficients of in each of for an INMA(1) model ................ 133 Table 6-4 Parameters of the over-lead-time-aggregated INARMA(p,q) model ............ 135 Table 7-1 Range of autoregressive and moving average parameters ............................ 142 Table 7-2 Range of INARMA parameters studied in the literature ............................... 142 Table 7-3 Parameter space for the selected INARMA models ...................................... 143 Table 8-1 MSE of YW, CLS and CML estimates for INAR(1) series when ...... 152 Table 8-2 Accuracy of YW and CLS estimates for INAR(1) series ............................... 153 Table 8-3 Accuracy of YW and CLS estimates for INMA(1) series ............................... 154 Table 8-4 Accuracy of YW and CLS estimates for INARMA(1,1) series ...................... 155 Table 8-5 One-step ahead forecast error comparison (YW, CLS and CML) for INAR(1) series .............................................................................................................................. 156 Table 8-6 One-step ahead forecast error comparison (YW and CLS) for INMA(1) series ....................................................................................................................................... 157 Table 8-7 One-step ahead forecast error comparison (YW and CLS) for INARMA(1,1) series .............................................................................................................................. 157 Table 8-8 Three-step ahead forecast error comparison (YW and CLS) for INAR(1) series ....................................................................................................................................... 158 Table 8-9 Six-step ahead forecast error comparison (YW and CLS) for INAR(1) series 159 Table 8-10 Three-step ahead forecast error comparison (YW and CLS) for INMA(1) series ....................................................................................................................................... 159 tY 11 ? ?? l jjtY }{ ijkt Z ? 1 1 ? ?? l jjtY }{ ijkt Z ? 1 1 ? ?? l jjtY }{ 24?n xiii Table 8-11 Six-step ahead forecast error comparison (YW and CLS) for INMA(1) series ....................................................................................................................................... 159 Table 8-12 Three-step ahead forecast error comparison (YW and CLS) for INARMA(1,1) series .............................................................................................................................. 160 Table 8-13 Six-step ahead forecast error comparison (YW and CLS) for INARMA(1,1) series .............................................................................................................................. 160 Table 8-14 The advantage of SBA over Croston for and all points in time ......... 163 Table 8-15 The advantage of SBA over Croston for and all points in time ......... 163 Table 8-16 One-step ahead for INARMA(0,0) series (known order) .............................................................................................................................. 167 Table 8-17 One-step ahead for INMA(1) series (known order) 168 Table 8-18 One-step ahead with smoothing parameter 0.2 for INAR(1) series (known order) ........................................................................................ 169 Table 8-19 One-step ahead with smoothing parameter 0.5 for INAR(1) series (known order) ........................................................................................ 169 Table 8-20 One-step ahead with smoothing parameter 0.2 for INARMA(1,1) series (known order) ............................................................................. 170 Table 8-21 One-step ahead with smoothing parameter 0.5 for INARMA(1,1) series (known order) .............................................................................. 170 Table 8-22 Three-step ahead for INARMA(0,0) series (known order) .............................................................................................................................. 171 Table 8-23 Three-step ahead for INMA(1) series (known order) ....................................................................................................................................... 171 Table 8-24 Three-step ahead with smoothing parameter 0.2 for INAR(1) series (known order) ........................................................................................ 171 Table 8-25 Three-step ahead with smoothing parameter 0.5 for INAR(1) series (known order) ........................................................................................ 172 Table 8-26 Three-step ahead with smoothing parameter 0.2 for INARMA(1,1) series (known order) ............................................................................. 172 Table 8-27 Three-step ahead with smoothing parameter 0.5 for INARMA(1,1) series (known order) .............................................................................. 173 Table 8-28 The percentage of correct identification for INARMA(0,0) series................ 175 Table 8-29 The percentage of correct identification for INAR(1) series ......................... 175 Table 8-30 Accuracy of INAR(1) forecasts for Ljung-Box and AIC identification procedures ...................................................................................................................... 177 Table 8-31 Accuracy of INAR(1) forecasts when the order is known ............................ 178 Table 8-32 Accuracy of forecasts with identification and all-INAR(1) for INARMA(0,0) series .............................................................................................................................. 179 Table 8-33 The percentage of correct identification for INARMA(0,0) series................ 181 20.?? 50.?? BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / xiv Table 8-34 The percentage of correct identification for INAR(1) series ......................... 181 Table 8-35 The percentage of correct identification for INMA(1) series ........................ 182 Table 8-36 The percentage of correct identification for INARMA(1,1) series................ 182 Table 8-37 Accuracy of INAR(1) forecasts for one-stage and two-stage identification procedures ...................................................................................................................... 184 Table 8-38 Accuracy of INMA(1) forecasts for one-stage and two-stage identification procedures ...................................................................................................................... 185 Table 8-39 Accuracy of INARMA(1,1) forecasts for one-stage and two-stage identification procedures ...................................................................................................................... 186 Table 8-40 Accuracy of INAR(1) forecasts when the order in known ............................ 187 Table 8-41 Accuracy of INMA(1) forecasts when the order in known ........................... 187 Table 8-42 Accuracy of INARMA(1,1) forecasts when the order in known................... 187 Table 8-43 Accuracy of forecasts with identification and all-INARMA(1,1) for INARMA(0,0) series ...................................................................................................... 189 Table 8-44 Accuracy of forecasts with identification and all-INARMA(1,1) for INAR(1) series .............................................................................................................................. 189 Table 8-45 Accuracy of forecasts with identification and all-INARMA(1,1) for INMA(1) series .............................................................................................................................. 190 Table 8-46 for INARMA(0,0) series (unknown order) ........... 191 Table 8-47 for INMA(1) series (unknown order) .................... 191 Table 8-48 with smoothing parameter 0.2 for INARMA(1,1) series (unknown order) ............................................................................................................. 192 Table 8-49 with smoothing parameter 0.5 for INARMA(1,1) series (unknown order) ............................................................................................................. 192 Table 8-50 BenchmarkINARMA MSEMSE / of lead-time forecasts for INARMA(0,0) series .............................................................................................................................. 194 Table 8-51 of lead-time forecasts for INMA(1) series 194 Table 8-52 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series .................................................................................... 194 Table 8-53 BenchmarkINARMA MSEMSE / of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series .................................................................................... 195 Table 8-54 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series ........................................................................... 195 Table 8-55 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series ........................................................................... 196 Table 8-56 of lead-time forecasts for INARMA(0,0) series .............................................................................................................................. 196 Table 8-57 of lead-time forecasts for INMA(1) series 197 BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / )( 3?l BenchmarkINARMA MSEMSE / )( 3?l BenchmarkINARMA MSEMSE / )( 3?l )( 3?l BenchmarkINARMA MSEMSE / )( 3?l BenchmarkINARMA MSEMSE / )( 3?l BenchmarkINARMA MSEMSE / )( 6?l BenchmarkINARMA MSEMSE / )( 6?l xv Table 8-58 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series .................................................................................... 197 Table 8-59 of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series .................................................................................... 197 Table 8-60 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series ........................................................................... 198 Table 8-61 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series ........................................................................... 198 Table 9-1 Comparing INAR(1) with benchmarks for all points in time (16000 series) 208 Table 9-2 Comparing INAR(1) with benchmarks for issue points (16000 series) ........ 209 Table 9-3 Comparing INAR(1) with benchmarks for all points in time (3000 series) .. 210 Table 9-4 Comparing INAR(1) with benchmarks for issue points (3000 series) .......... 211 Table 9-5 Comparing INARMA(1,1) with benchmarks for all points in time (16000 series) ............................................................................................................................. 212 Table 9-6 Comparing INARMA(1,1) with benchmarks for issue points (16000 series) ....................................................................................................................................... 212 Table 9-7 Comparing INARMA(1,1) with benchmarks for all points in time (3000 series) ............................................................................................................................. 213 Table 9-8 Comparing INARMA(1,1) with benchmarks for issue points (3000 series) 213 Table 9-9 The effect of identification on INARMA forecasts for all points in time (16000 series) ............................................................................................................................. 214 Table 9-10 The effect of identification on INARMA forecasts for issue points (16000 series) ............................................................................................................................. 215 Table 9-11 The effect of identification on INARMA forecasts for all points in time (3000 series) ............................................................................................................................. 216 Table 9-12 The effect of identification on INARMA forecasts for issue points (3000 series) ............................................................................................................................. 216 Table 9-13 Only INARMA(0,0) series for all points in time (3000 series) .................. 218 Table 9-14 Only INARMA(0,0) series for issue points (3000 series) ........................... 218 Table 9-15 Only INAR(1) series for all points in time (3000 series) ............................ 219 Table 9-16 Only INAR(1) series for issue points (3000 series) .................................... 219 Table 9-17 Only INMA(1) series for all points in time (3000 series) ........................... 220 Table 9-18 Only INMA(1) series for issue points (3000 series) ................................... 220 Table 9-19 Only INARMA(1,1) series for all points in time (3000 series) .................. 221 Table 9-20 Only INARMA(1,1) series for issue points (3000 series) ........................... 222 Table 9-21 Three-step-ahead YW-INAR(1) for all points in time (16000 series) ........ 223 Table 9-22 Three-step-ahead YW-INAR(1) for issue points (16000 series)................. 223 Table 9-23 Six-step-ahead YW-INAR(1) for all points in time (16000 series) ............ 224 BenchmarkINARMA MSEMSE / )( 6?l BenchmarkINARMA MSEMSE / )( 6?l BenchmarkINARMA MSEMSE / )( 6?l BenchmarkINARMA MSEMSE / )( 6?l xvi Table 9-24 Six-step-ahead YW-INAR(1) for issue points (16000 series) .................... 224 Table 9-25 Three-step-ahead YW-INAR(1) for all points in time (3000 series) .......... 225 Table 9-26 Three-step-ahead YW-INAR(1) for issue points (3000 series)................... 225 Table 9-27 Six-step-ahead YW-INAR(1) for all points in time (3000 series) .............. 226 Table 9-28 Six-step-ahead YW-INAR(1) for issue points (3000 series) ...................... 226 Table 9-29 The estimation and performance periods .................................................... 226 Table 9-30 The forecasting accuracy for all points in time for case 1 .......................... 227 Table 9-31 The forecasting accuracy for issue points for case 1 ................................... 227 Table 9-32 The forecasting accuracy for all points in time for case 2 .......................... 227 Table 9-33 The forecasting accuracy for issue points for case 2 ................................... 227 Table 9-34 The forecasting accuracy for all points in time for case 3 .......................... 228 Table 9-35 The forecasting accuracy for issue points for case 3 ................................... 228 Table 9-36 Lead-time forecasts for INAR(1) series for all points in time (16000 series) ............................................................................................................................. 230 Table 9-37 Lead-time forecasts for INAR(1) series for issue points (16000 series) ....................................................................................................................................... 230 Table 9-38 Lead-time forecasts for INAR(1) series for all points in time (16000 series) ............................................................................................................................. 230 Table 9-39 Lead-time forecasts for INAR(1) series for issue points (16000 series) ....................................................................................................................................... 231 Table 9-40 Lead-time forecasts for INAR(1) series for all points in time (3000 series) ............................................................................................................................. 231 Table 9-41 Lead-time forecasts for INAR(1) series for issue points (3000 series) ....................................................................................................................................... 232 Table 9-42 Lead-time forecasts for INAR(1) series for all points in time (3000 series) ............................................................................................................................. 232 Table 9-43 Lead-time forecasts for INAR(1) series for issue points (3000 series) ....................................................................................................................................... 233 Table 9-44 Lead-time forecasts for all-INAR(1) series for all points in time (16000 series) ................................................................................................................ 235 Table 9-45 Lead-time forecasts for all-INAR(1) series for issue points (16000 series) ............................................................................................................................. 235 Table 9-46 Lead-time forecasts for all-INAR(1) series for all points in time (16000 series) ................................................................................................................ 235 Table 9-47 Lead-time forecasts for all-INAR(1) series for issue points (16000 series) ............................................................................................................................. 236 Table 9-48 Lead-time forecasts for all-INAR(1) series for all points in time (3000 series) ............................................................................................................................. 236 )( 3?l )( 3?l )( 6?l )( 6?l )( 3?l )( 3?l )( 6?l )( 6?l )( 3?l )( 3?l )( 6?l )( 6?l )( 3?l xvii Table 9-49 Lead-time forecasts for all-INAR(1) series for issue points (3000 series) ............................................................................................................................. 237 Table 9-50 Lead-time forecasts for all-INAR(1) series for all points in time (3000 series) ............................................................................................................................. 237 Table 9-51 Lead-time forecasts for all-INAR(1) series for issue points (3000 series) ............................................................................................................................. 238 Table 6.A-1 Coefficients of in each of for an INAR(2) model .................. 294 Table 6.A-2 Coefficients of in each of for an INAR(2) model ................ 295 Table 6.A-3 Coefficients of in each of for an INAR(2) model ............. 296 Table 6.B-1 Coefficients of in each of for an INARMA(1,2) model ......... 298 Table 6.B-2 Coefficients of in each of for an INARMA(1,2) model .... 299 Table 6.C-1 Parameters of the over-lead-time-aggregated INARMA(p,q) model ........ 302 Table 8.A-1 MSE of YW and CLS estimates of for an INAR(1) process ................ 303 Table 8.A-2 Comparison of YW and CLS estimates of for an INAR(1) process ..... 303 Table 8.A-3 Comparison of YW and CLS estimates of for an INMA(1) process ... 304 Table 8.A-4 Comparison of YW and CLS estimates of for an INMA(1) process .... 304 Table 8.A-5 Comparison of YW and CLS estimates of for an INARMA(1,1) process ....................................................................................................................................... 305 Table 8.A-6 Comparison of YW and CLS estimates of for an INARMA(1,1) process ....................................................................................................................................... 305 Table 8.A-7 Comparison of YW and CLS estimates of for an INARMA(1,1) process ....................................................................................................................................... 305 Table 8.B-1 Forecast error comparison (YW and CLS) for INAR(1) series ................... 306 Table 8.B-2 Forecast error comparison (YW and CLS) for INMA(1) series .................. 306 Table 8.B-3 Comparison error comparison (YW and CLS) for INARMA(1,1) series .... 307 Table 8.C-1 MSE of Croston and SBA with smoothing parameter 0.2 for INAR(1) series ....................................................................................................................................... 308 Table 8.C-2 MSE of Croston and SBA with smoothing parameter 0.5 for INAR(1) series ....................................................................................................................................... 308 Table 8.C-3 MSE of Croston and SBA with smoothing parameter 0.2 for INMA(1) series .............................................................................................................................. 309 Table 8.C-4 MSE of Croston and SBA with smoothing parameter 0.5 for INMA(1) series .............................................................................................................................. 309 Table 8.C-5 MSE of Croston and SBA with smoothing parameter 0.2 for INARMA(1,1) series .............................................................................................................................. 310 Table 8.C-6 MSE of Croston and SBA with smoothing parameter 0.5 for INARMA(1,1) series .............................................................................................................................. 310 )( 3?l )( 6?l )( 6?l tY 11 ? ?? l jjtY }{ 1?tY 11 ? ?? l jjtY }{ ijkt Z ? 1 1 ? ?? l jjtY }{ tY 11 ? ?? l jjtY }{ ijkt Z ? 1 1 ? ?? l jjtY }{ ? ? ? ? ? ? ? xviii Table 8.D-1 Forecast accuracy for all points in time and issue points for INAR(1) series (known order) ................................................................................................................. 311 Table 8.D-2 Forecast accuracy for all points in time and issue points for INMA(1) series (known order) ................................................................................................................. 312 Table 8.D-3 Forecast accuracy for all points in time and issue points for INARMA(1,1) series (known order) ....................................................................................................... 313 Table 8.E-1 for INARMA(0,0) series (known order) ......... 314 Table 8.E-2 for INMA(1) series (known order) .................. 314 Table 8.E-3 for INAR(1) series (known order)................... 315 Table 8.E-4 for INARMA(1,1) series (known order) ......... 316 Table 8.F-1 Six-step ahead for INARMA(0,0) series (known order) .............................................................................................................................. 317 Table 8.F-2 Six-step ahead for INMA(1) series (known order) 317 Table 8.F-3 Six-step ahead with smoothing parameter 0.2 for INAR(1) series (known order) ........................................................................................ 318 Table 8.F-4 Six-step ahead with smoothing parameter 0.5 for INAR(1) series (known order) ........................................................................................ 318 Table 8.F-5 Six-step ahead with smoothing parameter 0.2 for INARMA(1,1) series (known order) ............................................................................. 319 Table 8.F-6 Six-step ahead with smoothing parameter 0.5 for INARMA(1,1) series (known order) .............................................................................. 319 Table 8.G-1 Accuracy of forecasts by all-INAR(1) and all-INARMA(1,1) approaches for INAR(1) series ............................................................................................................... 320 Table 8.G-2 Accuracy of forecasts by all-INAR(1) and all-INARMA(1,1) approaches for INMA(1) series .............................................................................................................. 321 Table 8.G-3 Accuracy of forecasts by all-INAR(1) and all-INARMA(1,1) approaches for INARMA(1,1) series ...................................................................................................... 322 Table 8.H-1 for INARMA(0,0) series (unknown order) ..... 323 Table 8.H-2 for INMA(1) series (unknown order) ............. 323 Table 8.H-3 for INARMA(1,1) series (unknown order) ..... 324 Table 8.I-1 of lead-time forecasts for INARMA(0,0) series .............................................................................................................................. 325 Table 8.I-2 of lead-time forecasts for INMA(1) series ....................................................................................................................................... 325 Table 8.I-3 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series .................................................................................... 326 Table 8.I-4 of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series .................................................................................... 326 BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 3?l xix Table 8.I-5 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series ........................................................................... 326 Table 8.I-6 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series ........................................................................... 327 Table 8.I-7 of lead-time forecasts for INARMA(0,0) series .............................................................................................................................. 327 Table 8.I-8 of lead-time forecasts for INMA(1) series ....................................................................................................................................... 327 Table 8.I-9 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series .................................................................................... 328 Table 8.I-10 of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series .................................................................................... 328 Table 8.I-11 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series ........................................................................... 328 Table 8.I-12 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series ........................................................................... 329 Table 9.A-1 Only INARMA(0,0) series for all points in time (16000 series) ............... 330 Table 9.A-2 Only INARMA(0,0) series for issue points (16000 series) ....................... 331 Table 9.A-3 Only INAR(1) series for all points in time (16000 series) ........................ 331 Table 9.A-4 Only INAR(1) series for issue points (16000 series) ................................ 331 Table 9.A-5 Only INMA(1) series for all points in time (16000 series) ....................... 332 Table 9.A-6 Only INMA(1) series for issue points (16000 series) ............................... 332 Table 9.A-7 Only INARMA(1,1) series for all points in time (16000 series) ............... 332 Table 9.A-8 Only INARMA(1,1) series for issue points (16000 series) ....................... 333 Table 9.B-1 Three-step-ahead INARMA(0,0) for all points in time (16000 series) ..... 334 Table 9.B-2 Three-step-ahead INARMA(0,0) for issue points (16000 series) ............. 334 Table 9.B-3 Six-step-ahead INARMA(0,0) for all points in time ( 16000 series) ........ 334 Table 9.B-4 Six-step-ahead INARMA(0,0) for issue points (16000 series) ................. 335 Table 9.B-5 Three-step-ahead YW-INMA(1) for all points in time (16000 series) ...... 335 Table 9.B-6 Three-step-ahead YW-INMA(1) for issue points (16000 series) .............. 335 Table 9.B-7 Six-step-ahead YW-INMA(1) for all points in time ( 16000 series) ......... 335 Table 9.B-8 Six-step-ahead YW-INMA(1) for issue points (16000 series) .................. 336 Table 9.B-9 Three-step-ahead INARMA(1,1) for all points in time (16000 series) ..... 336 Table 9.B-10 Three-step-ahead INARMA(1,1) for issue points (16000 series) ........... 336 Table 9.B-11 Six-step-ahead INARMA(1,1) for all points in time ( 16000 series) ...... 336 Table 9.B-12 Six-step-ahead INARMA(1,1) for issue points (16000 series) ............... 337 Table 9.B-13 Three-step-ahead INARMA(0,0) for all points in time (3000 series) ..... 337 Table 9.B-14 Three-step-ahead INARMA(0,0) for issue points (3000 series) ............. 337 BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l xx Table 9.B-15 Six-step-ahead INARMA(0,0) for all points in time ( 3000 series) ........ 337 Table 9.B-16 Six-step-ahead INARMA(0,0) for issue points (3000 series) ................. 338 Table 9.B-17 Three-step-ahead YW-INMA(1) for all points in time (3000 series) ...... 338 Table 9.B-18 Three-step-ahead YW-INMA(1) for issue points (3000 series) .............. 338 Table 9.B-19 Six-step-ahead YW-INMA(1) for all points in time ( 3000 series) ......... 338 Table 9.B-20 Six-step-ahead YW-INMA(1) for issue points (3000 series) .................. 339 Table 9.B-21 Three-step-ahead CLS-INARMA(1,1) for all points in time (3000 series) ....................................................................................................................................... 339 Table 9.B-22 Three-step-ahead CLS-INARMA(1,1) for issue points (3000 series) ..... 339 Table 9.B-23 Six-step-ahead CLS-INARMA(1,1) for all points in time (3000 series) 339 Table 9.B-24 Six-step-ahead CLS-INARMA(1,1) for issue points (3000 series) ......... 340 Table 9.C-1 Lead-time forecasts for INARMA(0,0) series for all points in time (16000 series) ................................................................................................................ 341 Table 9.C-2 Lead-time forecasts for INARMA(0,0) series for issue points (16000 series) ............................................................................................................................. 341 Table 9.C-3 Lead-time forecasts for INARMA(0,0) series for all points in time (16000 series) ................................................................................................................ 341 Table 9.C-4 Lead-time forecasts for INARMA(0,0) series for issue points (16000 series) ............................................................................................................................. 342 Table 9.C-5 Lead-time forecasts for INMA(1) series for all points in time (16000 series) ............................................................................................................................. 342 Table 9.C-6 Lead-time forecasts for INMA(1) series for issue points (16000 series) ............................................................................................................................. 342 Table 9.C-7 Lead-time forecasts for INMA(1) series for all points in time (16000 series) ............................................................................................................................. 343 Table 9.C-8 Lead-time forecasts for INMA(1) series for issue points (16000 series) ............................................................................................................................. 343 Table 9.C-9 Lead-time forecasts for INARMA(1,1) series for all points in time (16000 series) ................................................................................................................ 343 Table 9.C-10 Lead-time forecasts for INARMA(1,1) series for issue points (16000 series) ................................................................................................................ 344 Table 9.C-11 Lead-time forecasts for INARMA(1,1) series for all points in time (16000 series) ................................................................................................................ 344 Table 9.C-12 Lead-time forecasts for INARMA(1,1) series for issue points (16000 series) ................................................................................................................ 344 Table 9.C-13 Lead-time forecasts for INARMA(0,0) series for all points in time (3000 series) .................................................................................................................. 345 Table 9.C-14 Lead-time forecasts for INARMA(0,0) series for issue points (3000 series) ............................................................................................................................. 345 )( 3?l )( 3?l )( 6?l )( 6?l )( 3?l )( 3?l )( 6?l )( 6?l )( 3?l )( 3?l )( 6?l )( 6?l )( 3?l )( 3?l xxi Table 9.C-15 Lead-time forecasts for INARMA(0,0) series for all points in time (3000 series) .................................................................................................................. 345 Table 9.C-16 Lead-time forecasts for INARMA(0,0) series for issue points (3000 series) ............................................................................................................................. 345 Table 9.C-17 Lead-time forecasts for INMA(1) series for all points in time (3000 series) ............................................................................................................................. 346 Table 9.C-18 Lead-time forecasts for INMA(1) series for issue points (3000 series) ............................................................................................................................. 346 Table 9.C-19 Lead-time forecasts for INMA(1) series for all points in time (3000 series) ............................................................................................................................. 346 Table 9.C-20 Lead-time forecasts for INMA(1) series for issue points (3000 series) ............................................................................................................................. 347 Table 9.C-21 Lead-time forecasts for INARMA(1,1) series for all points in time (3000 series) .................................................................................................................. 347 Table 9.C-22 Lead-time forecasts for INARMA(1,1) series for issue points (3000 series) ............................................................................................................................. 347 Table 9.C-23 Lead-time forecasts for INARMA(1,1) series for all points in time (3000 series) .................................................................................................................. 348 Table 9.C-24 Lead-time forecasts for INARMA(1,1) series for issue points (3000 series) ............................................................................................................................. 348 )( 6?l )( 6?l )( 3?l )( 3?l )( 6?l )( 6?l )( 3?l )( 3?l )( 6?l )( 6?l xxii List of abbreviations ACF Autocorrelation Function AE Absolute Error ARIMA Autoregressive Integrated Moving Average CLS Conditional Least Squares CML Conditional Maximum Likelihood DARMA Discrete Autoregressive Moving Average DSD Discrete Self-Decomposable EWMA Exponentially Weighted Moving Average GMM General Method of Moments IDF Intermittent Demand Forecasting INARMA Integer Autoregressive Moving Average LTD Lead Time Demand MAD Mean Absolute Deviation MAPE Mean Absolute Percentage Error MASE Mean Absolute Scaled Error MC Modified Croston MCMC Markov Chain Monte Carlo ME Mean Error MSE Mean Square Error ML Maximum Likelihood PACF Partial Autocorrelation Function PB Percentage Better RGRMSE Relative Geometric Root Mean Square Error SBA Syntetos-Boylan Approximation SBJ Shale-Boylan-Johnston SES Single (or simple) Exponential Smoothing SMA Simple Moving Average YW Yule Walker ZF Zero Forecast M.Mohammadipour, 2009, Chapter 1 1 Chapter 1 INTRODUCTION This chapter lays the foundations for this thesis and provides an overall perspective for this PhD research. It introduces the research by means of an overview of the project. The business context and research background are described. The research problem and research questions are then defined and the methodology is briefly outlined. All the above topics are explained in detail by a step-by-step process through chapters 2 to 9. However, the introductory chapter is intended to outline the research through a summary of the research background and problems, expected results, and designated methodology. The structure of chapter 1 and sequences of sections are shown in Figure 1-1. M.Mohammadipour, 2009, Chapter 1 2 Figure ?1-1 The structure of Chapter 1 1.1 Introduction To attain a unified understanding of related concepts in this PhD thesis, a brief description of the key terms and phrases used in the study is provided in this section. These are the definitions that will be adopted in this thesis. More discussion will follow in later chapters. Time series Bowerman et al. (2005) have defined a time series as ?a chronological sequence of observations on a particular variable? (p.4). Demand of a product over time and inventory level for a product over time are examples of time series. One of the most general classes of models for forecasting a time series is the class of autoregressive integrated moving average (ARIMA) models (Box et al., 1994). ARIMA models are based on adding linear combinations of lags of the differenced series and/or lags of the forecast errors to the prediction equation, as needed to remove any last traces of autocorrelation from the forecast errors. Time series of counts Time series of counts is defined by McKenzie (2003) as ?counts of events, objects or 1.1 Introduction 1.2 Research Overview 1.5 Research Problem 1.3 Business Context 1.6 Research Methodology 1.7 Thesis Structure 1.4 Research Background M.Mohammadipour, 2009, Chapter 1 3 individuals in consecutive intervals or at consecutive points in time?. Integer autoregressive moving average (INARMA) models have been proposed for forecasting time series of counts, and have received a certain amount of attention over the last 25 years. These models are explained in detail in chapter 3. Intermittent series Intermittent series are time series of non-negative integer values where some values are zero (Shenstone and Hyndman, 2005). A common example is intermittent demand. Intermittent demand should be distinguished from lumpy demand where the nonzero values are highly variable. Methods for intermittent demand forecasting (IDF) are reviewed in chapter 2. 1.2 Research Overview This PhD concentrates on modelling intermittent demand by integer autoregressive moving average (INARMA) processes and proposing a forecasting method based on such processes. The study first focuses on stochastic characteristics of INARMA processes, including finding the second unconditional moment and the autocorrelation function (ACF) and partial autocorrelation function (PACF) structure. Lead time forecasts are also developed for INARMA models. INARMA models are then used for intermittent demand forecasting. The results are compared to some benchmark methods in the literature, namely Croston's method (Croston, 1972), Syntetos-Boylan Approximation (SBA) (Syntetos and Boylan, 2005), and Shale-Boylan-Johnston (SBJ) (Shale et al., 2006). The rationale for this choice of benchmarks is given in chapter 2. 1.3 Business Context Accurate demand forecasting is a significant concern for many organizations. It lays a foundation for every part of inventory management. Various forecasting models have been developed to incorporate different components of demand such as trend and seasonality. A class of demand called intermittent demand exists in which some M.Mohammadipour, 2009, Chapter 1 4 periods have no demand at all. This is especially the case for service parts and capital goods (Willemain et al., 2004). Examples are commonly found in the aerospace, automotive, computer components, electronics, and industrial machinery industries. In general, all companies that must stock spare parts face intermittent demand. Intermittent time series are also common in business and economic data. There are many cases for which non-intermittent data is intermittent at lower levels of data disaggregation (for example, greater frequency or smaller geography). The items with intermittent demand are not rare and in fact they constitute the majority of items held by many stockists (Johnston et al., 2003). These items are also important from a financial point of view. In the U.S. alone, service parts management grew to a $700 billion business sector in 2001 (Patton and Steele, 2003). The fact that, in many cases, service parts face a high risk of obsolescence makes accurate forecasting for such items even more important. Lower stock-holding costs and higher service levels are the outcomes of more accurate forecasts. INARMA models have had applications in a wide area including medical science (Franke and Seligmann, 1993; Cardinal et al., 1999), economics (B?ckenholt, 1999; Br?nn?s and Hellstr?m, 2001; Freeland and McCabe, 2004b) and service industries (Br?nn?s et al., 2002). However, the performance of these models in an intermittent demand context is yet to be tested. This testing is to be conducted in this research as a new application for INARMA models. Although the empirical analysis of this thesis focuses on intermittent demand data, the theoretical and simulation findings can generally be applied to any time series with low non-negative integer-valued data. 1.4 Research Background Many companies use single exponential smoothing (SES) for intermittent demand forecasting (Teunter and Sani, 2009) but Croston (1972) shows that this can result in biased forecasts and excessive stock levels. He proposes a method based on separate exponential smoothing estimates of demand size and interval between demands, M.Mohammadipour, 2009, Chapter 1 5 which he claims to be unbiased. However, Syntetos and Boylan (2001) show that Croston?s method is positively biased. Some methods have been suggested to reduce this bias (Lev?n and Segerstedt, 2004; Syntetos and Boylan, 2005). While it is confirmed that the correction by Syntetos and Boylan does reduce the bias of Croston?s method, the modification by Lev?n and Segerstedt is even more biased than Croston?s method (Teunter and Sani, 2009). In parallel with these studies, some authors focus on finding a model underlying Croston?s method (Shenstone and Hyndman, 2005; Snyder and Beaumont, 2007). This is especially useful to find the distribution and prediction intervals of the forecasts. However, none of these studies has yet demonstrated an underlying model for which Croston?s method is optimal. It has been suggested that the focus should now be moved to stationary models for IDF such as time series models for counts (Shenstone and Hyndman, 2005). Time series models for counts occur as counts of individuals (e.g. the number of people in a queue waiting to receive a service at a certain moment) or events (e.g. the number of accidents in a firm each three months). If these discrete variates are large numbers, they could be approximated by continuous variates; otherwise, special models should be used. A class of models for count data has been developed, namely the integer autoregressive moving average (INARMA) models. These models were originally introduced in the 1980s (McKenzie, 1985; Al-Osh and Alzaid, 1987) and it will be shown in chapter 3 that they are analogous to well-known ARMA models (Box et al., 1994). Intermittent demand data belong to a broader class called count data. Although count data frequently occur in many industries, not many stochastic models have been developed for them (Gooijer and Hyndman, 2006). As one of these few models, integer autoregressive moving average models have recently received much attention. This PhD thesis aims to bridge the gap between models for counts and intermittent demand forecasting, focusing specifically on INARMA models. In doing so, some issues such as identification, estimation of parameters and lead time forecasting need to be addressed. Identification of the autoregressive and moving M.Mohammadipour, 2009, Chapter 1 6 average orders of INARMA models has been mainly conducted in the literature using the autocorrelation function (ACF) and the partial autocorrelation function (PACF) (Jung and Tremayne, 2006a; Zheng et al., 2006; Zhu and Joe, 2006; Bu and McCabe, 2008). To our knowledge, the ACF of the general INARMA(p,q) process has not been looked at before. Therefore, we establish the unconditional variance and the ACF of such processes. The conditional expected value of the over-lead-time aggregated process are also obtained. The accuracy of h-step-ahead and lead time INARMA forecasts are compared to those obtained by the benchmark methods mentioned in section 1.2. In doing so, the difficulties of forecasting intermittent demand are borne in mind, particularly if the length of data history is short or the data is sparse. 1.5 Research Problem 1.5.1 Initial Problem The main problem addressed in this research is as follows: o In the context of intermittent demand, is there any benefit in modelling and forecasting the demand using INARMA models, in terms of forecast accuracy, compared to simpler methods? 1.5.2 Research Questions Based upon the above initial research problem, the four detailed questions for the research are as follows: I. How can the appropriate integer autoregressive moving average (INARMA) model be identified for a time series of counts? Different methods have been suggested for identification of ARIMA processes including: using the sample autocorrelation function (SACF) and the sample partial autocorrelation function (SPACF), and using a penalty function such as the Akaike information criterion (AIC) or the Bayesian information criterion M.Mohammadipour, 2009, Chapter 1 7 (BIC). This research focuses on finding the ACF and PACF structure of INARMA processes. But, using ACF and PACF needs a visual check. Identification based on these functions has not been automated for INARMA models as it has been for ARMA models (e.g. in AutoBox). Therefore automated methods should be used. It has been suggested in the literature that data should be first tested for any serial dependence. As a result, two identification procedures are compared in this research: a two-stage procedure first uses a test of serial dependence to distinguish between an independent and identically distributed (i.i.d.) process and other INARMA processes and then the AIC is used to select among other possible INARMA processes. A one-stage procedure does not have the first step and only uses the AIC. However, the correct model might not be identified at all times. In such cases, it is of interest to find the impact that misidentification has on forecast accuracy. II. How can the parameters of integer autoregressive moving average (INARMA) models be estimated? The performance of different estimation methods are to be tested in terms of both the accuracy of parameter estimates and their impact on forecast accuracy. III. How can an INARMA process be forecasted over a lead time? One of the application areas of lead-time aggregation is in the inventory control field where there is a lead time between placing an order by a manufacturer and receiving it from its supplier. The manufacturer has to place an order to cover the demand over the lead time and, therefore, the lead-time demand has to be forecasted. The aggregated INARMA(p,q) process over a lead time and its conditional expected value are found. The latter is then used as the lead time forecast. IV. Do INARMA models provide more accurate forecasts for intermittent demand than non-optimal smoothing-based methods? This research has suggested using INARMA models to forecast intermittent demand. The accuracy of forecasts provided by INARMA methods then has to be compared with some of the methods that have been used in the literature of M.Mohammadipour, 2009, Chapter 1 8 intermittent demand forecasting. As this research solely focuses on demand forecasting and not inventory control, forecast-accuracy metrics will be used rather than accuracy-implication metrics (measures that analyze the effect of forecasting methods on inventory performance measures (Boylan and Syntetos, 2006)). As previously mentioned, the methods that will be used for comparison purposes are Croston's method (Croston, 1972), the Syntetos-Boylan Approximation (Syntetos and Boylan, 2005), and the method of Shale-Boylan- Johnston (Shale et al., 2006). 1.6 Research Methodology This section will try to sufficiently clarify the ?philosophy?, ?approach? and ?strategy? of this study to demonstrate how the expected results in this PhD may be achieved. This research follows the ?positivism? philosophy where the research approach will be ?deductive?. Accordingly, based upon the detailed hypotheses and the required level of generalisation, ?simulation? and then ?empirical study? is believed to be the most suitable strategy for this research. To explain the research methodology of this study, the approach of the ?research process onion? as termed by Saunders et al. (2003) is followed. On this basis, an appropriate research philosophy, research approach, and research strategy are proposed for the study. Consistent with the research strategy, source(s) of data are identified for analysis. The structure of this section is shown in Figure 1-2. Figure ?1-2 The structure of the research methodology 1.6.1 Research Philosophy 1.6.2 Research Approach 1.6.3 Research Strategy 1.6.3.1 Mathematical Analysis 1.6.3.2 Simulation 1.6.3.3 Empirical Study M.Mohammadipour, 2009, Chapter 1 9 1.6.1 Research Philosophy Taking into account the existing knowledge of the overall environment of forecasting and time series analysis, the ground philosophy of this research tends towards ?positivism? (rather than pure ?realism? or ?interpretivism?), i.e. the end product of the research can be law-like generalisations. The nature of intermittent data makes it difficult for humans to detect patterns for this type of data. This means that a positivistic deductive approach is a natural starting point. An interpretivistic approach may be more appropriate for investigating human adjustments to intermittent demand forecasts. 1.6.2 Research Approach Research approaches are classified into two main groups of ?deductive? and ?inductive?. While the former works on the basis of a clear understanding of the research theory and questions, the latter tries to find these through investigations with real world data (Saunders et al., 2003). Although some inductive studies have been done in the area of forecasting (e.g. Collopy and Armstrong, 1992; Adya et al., 2001), most of the studies in this field are deductive. A theory is developed first and a research strategy to test it is designed. This research is also based on developing a theoretical structure and testing the findings by simulation and empirical analysis. Hence, a deductive approach is followed in this PhD thesis. 1.6.3 Research Strategy A research strategy endeavours to plan the process of answering the research questions (section 1.5.2). Our strategy consists of three steps: mathematical analysis, simulation and empirical study. 1.6.3.1 Mathematical Analysis Although the introduction of INARMA models dates back to the 1980s, there are still M.Mohammadipour, 2009, Chapter 1 10 an increasing number of studies which investigate the statistical properties of these models. The mathematical analysis of this research aims at extending the theory of INARMA modelling making it more complete. The mathematical results of this PhD are provided in chapters 3 to 6. Some stochastic properties of the general INARMA(p,q) process have been obtained which are: the unconditional variance and the autocorrelation function (ACF) and partial autocorrelation function (PACF). The results of aggregation of an INARMA(p,q) process over lead time along with the conditional expected value of the aggregated process are established. These results are then used in chapters 8 and 9 to compare the performance of the INARMA method with some benchmark methods in forecasting intermittent demand. 1.6.3.2 Simulation Simulation will be used for the following reasons: ? to assess the percentage of theoretically generated INARMA time series that can be identified correctly ? to investigate the effect of misidentification on forecasting accuracy ? to compare the performance of different estimation methods ? to assess the sensitivity of the results (parameter estimates and forecasting accuracy) to the sparsity of data and the length of history ? to compare the INARMA forecasts with those of the benchmark methods It can be seen that simulation is essential for the first three objectives (and the part of the fourth objective relating to parameter estimates) because only for theoretically generated data, the order of the INARMA model and the parameters are known. For the remaining objectives, simulation is useful to gain additional insights. The design of the simulation study is explained in chapter 7. M.Mohammadipour, 2009, Chapter 1 11 1.6.3.3 Empirical Analysis The findings of this PhD thesis are to be tested on real empirical data to assess the practical validity and applicability of the main results of the study. As discussed, simulation helps us to test the accuracy of model identification, when we know that the intermittent demand follows an INARMA process with known order and parameters. However, this is not true for real data in that we do not have such information from the beginning. Therefore, empirical analysis would help us to test the applicability of the results in real situations. Although many studies focus on the statistical aspects of INARMA models, there are fewer studies regarding the application of these models (Jung and Tremayne, 2003). There is especially a lack of empirical testing of INARMA models on intermittent demand. The demand data series used in this PhD thesis are Royal Air Force (RAF) individual demand histories of 16,000 SKUs over a period of 6 years (monthly observations) and 3,000 series of intermittent demand for 24 periods (two years monthly series) from the automotive industry. 1.7 Thesis Structure Based on the position adumbrated in this chapter, this PhD thesis is structured as follows. Chapter 2 discusses different definitions and categorizations of intermittent demand, reviews methods for intermittent demand forecasting and the accuracy measures that can be used in this context. Chapter 3 briefly reviews different count models and introduces INARMA models. The stochastic properties of these models and their application in the literature are examined. The unconditional variance and ACF of the INARMA(p,q) process are obtained. Chapter 4 reviews different approaches for identification of the order of INARMA models. The PACF structure of INARMA models is also derived. Chapter 5 discusses methods for estimation of the parameters of these models. Chapter 6 investigates the forecasting of an INARMA process over lead time. Chapter 7 discusses the design of simulation, and the results of simulation experiments M.Mohammadipour, 2009, Chapter 1 12 are presented in Chapter 8. Chapter 9 assesses the validity of theoretical and simulation results on real intermittent demand data. Finally, the findings of this PhD research, the limitations and some potential future studies are reviewed in chapter 10. The structure of the thesis is shown in Figure 1-3. Figure ?1-3 The structure of the thesis Chapter 1 Introduction Chapter 2 Forecasting Intermittent Demand Chapter 5 Estimation in INARMA Models Chapter 3 Integer Autoregressive Moving Average Models Chapter 6 Forecasting in INARMA Models Chapter 7 Simulation Design Chapter 4 Identification in INARMA Models Chapter 8 Simulation Results Chapter 10 Conclusions and Further Research Chapter 9 Empirical Analysis M.Mohammadipour, 2009, Chapter 2 13 Chapter 2 FORECASTING INTERMITTENT DEMAND 2.1 Introduction This chapter aims to provide an overview of the literature on forecasting intermittent demand. This research focuses on integer autoregressive moving average (INARMA) models to address intermittent demand modelling and forecasting. As the focus of this research is on forecasting, we do not review the literature on inventory control for slow-moving items. The chapter is organized as follows. Intermittent demand is defined in detail in section 2.2. Methods for forecasting intermittent demand are then discussed in section 2.3. We start our review with Croston?s method (Croston, 1972) as the most M.Mohammadipour, 2009, Chapter 2 14 widely used approach in this field. Some variants of Croston?s method are then reviewed. Studies based on bootstrapping to forecast lead-time demand are also discussed. As this research is based on comparing different forecasting methods, the forecast accuracy measures need to be selected. Some accuracy measures cannot be used in an intermittent demand context. Section 2.4 determines these measures and classifies the measures that can be applied to intermittent series. A number of studies have compared different methods of forecasting intermittent demand. These studies are reviewed in section 2.5. The motives to use INARMA models for modelling and forecasting intermittent demand are discussed in section 2.6 and, finally, section 2.7 concludes the chapter. 2.2 Definition of Intermittent Demand Intermittent demand is defined by Silver et al. (1998) as ?infrequent in the sense that the average time between transactions is considerably larger than the unit period, the latter being the interval of forecast updating? (footnote, p.127). The main disadvantage of this definition is its impracticality, i.e. it does not define how long the average time between transactions should be for demand to be considered intermittent. The definition provided by Johnston and Boylan (1996) is more practical. They suggest that demand is intermittent when the mean inter-arrival time between demands is greater than 1.25 review intervals. This cut-off value is based on a comparison of Croston's method and Single Exponential Smoothing (SES) using simulated data. However, this definition also has its limitations: it is simulation based, it depends on specific methods, and it does not take into account the ?lumpiness? of the demand. Intermittent demand is often lumpy, which means that the variability among the nonzero values is high (Willemain et al., 2004). However, these two concepts should be distinguished. Shenstone and Hyndman (2005) introduce another practical definition: ?Data for intermittent demand items consist of time series of non-negative integer values where some values are zero?. It can be seen that this does not take into account the lumpiness of demand either. M.Mohammadipour, 2009, Chapter 2 15 There are two main categorization schemes for intermittent demand in the literature. The first approach, proposed by Williams (1984) and later revised by Eaves and Kingsman (2004), is based on second moment variability (transaction variability, demand size variability and lead-time variability). The other approach, suggested by Syntetos et al. (2005) is based on frequency of transactions and demand size variability. The classification approach by Eaves and Kingsman (2004) is a revision of the method of Williams (1984) who classified demand by decomposing the variance of lead-time demand into transaction variability, demand size variability and lead-time variability. This is shown in Table 2-1. Table ?2-1 The categorization scheme for intermittent demand data (Eaves and Kingsman, 2004) Lead-time demand component Demand pattern classification Transaction variability Demand size variability Lead-time variability Low Low Smooth Low High Irregular High Low Slow moving High High Low Mildly intermittent High High High Highly intermittent The main disadvantage of Eaves and Kingsman?s classification is its lack of practicality. Unlike Syntetos et al. (2005), they do not provide cut-off values for different classes. Syntetos et al. (2005) categorize demand based on the expected mean square error (MSE) of each forecasting method. They propose four categories of demand shown in Figure 2-1 which are: ?erratic?, ?lumpy?, ?smooth?, and ?intermittent?. Each of these demand classes are uniquely specified by two parameters: ? which is the average inter-demand interval, and ? which is the squared coefficient of variation of the demand when it occurs. The classification by Johnston and Boylan (1996) is based on comparing Croston's method with SES as methods widely used in forecasting software packages. Therefore, although it has the benefit of being practical, one can argue that it is not a comprehensive classification method that can be used for all forecasting methods. The categorization scheme of Syntetos et al. (2005) also includes the Syntetos- M.Mohammadipour, 2009, Chapter 2 16 Boylan Approximation (SBA) (explained in section 2.3.1.1) in addition to the other two methods. The main benefit of their scheme is again its practicality (because of determining the cut-off values) and that it is empirically validated. However, as the authors mentioned, the categories are based on only one forecast accuracy measure (MSE) (see section 2.4 for more information on accuracy measures). Figure ?2-1 The categorization scheme for intermittent demand data (Syntetos et al. 2005) The classification by Syntetos et al. (2005) distinguishes between intermittent and smooth demand to determine which method should be used in each situation. This is because the Syntetos-Boylan Approximation and SES methods are not universal, i.e. these methods are not appropriate for both categories of demand. This research aims at using integer autoregressive moving average (INARMA) models to model and forecast intermittent demand. This method can be used for both intermittent and non-intermittent data provided that the data is not lumpy. Therefore, we do not need to focus on a specific classification method for intermittence to find when the method should be used. We do need to follow a standard definition that is practical and universal (not limited to specific methods). Among the definitions that we reviewed before, only the one by Shenstone and Hyndman (2005) met all the above criteria and therefore is used for the purpose of this study. As a result, whenever a data series has at least one zero, it is considered as an intermittent data series, and an INARMA forecasting method can be used. However, the universality of the INARMA approach means that it can also be used Lumpy Erratic Intermittent Smooth ?= 0.49 ?= 1.31 M.Mohammadipour, 2009, Chapter 2 17 when all the observations are positive. As previously mentioned, although the classification by Syntetos et al. (2005) is practical, it is only based on specific forecasting methods. We will test this classification when the data is INARMA. INARMA forecasts will be compared to the best benchmark method based on this classification. The main reason for not using Eaves and Kingsman classification method is its lack of practicality and also because lead-time variability will not be studied in this thesis. As mentioned before, the other reason for the selection of Syntetos et al.?s classification is that two methods used in their classification (Croston?s method and SBA) will be used in this research for comparison reasons. This research focuses on using INARMA models with Poisson marginal distributions to model intermittent demand. Due to the properties of the Poisson distribution, lumpy demand cannot be modelled by these models. As will be explained in chapter 9, a filtering mechanism will be used to eliminate the lumpy series when dealing with empirical data. 2.3 Methods of Forecasting Intermittent Demand Inventories with intermittent demands are common in practice (Shenstone and Hyndman, 2005) and they create significant problems in the manufacturing and supply environment (Syntetos and Boylan, 2001). The accurate forecasting of demand is one of the most important issues of inventory management (Ghobbar and Friend, 2003). This is more difficult when demand has an intermittent nature (Willemain et al., 2004). Willemain et al. (2004) divide intermittent demand forecasting (IDF) methods into: ? simple statistical smoothing methods ? Croston?s method and its variants ? bootstrap methods from which we focus on the last two classes because, according to Croston (1972), the first class results in biased forecasts when applied immediately after a demand M.Mohammadipour, 2009, Chapter 2 18 occurrence. These methods have also been the subject of some comparison studies, including Teunter and Duncan (2009). These studies are reviewed in section 2.5. 2.3.1 Croston?s?Method The most widely used approach for forecasting intermittent demand is Croston?s method (Shenstone and Hyndman, 2005). Croston (1972) suggested that conventional forecasting methods like Simple Moving Averages (SMA) and Single Exponential Smoothing (SES) may not be appropriate for slow-moving items. The bias associated with SES is expressed in a quantitative form for the case in which forecasts are updated after a demand occurrence. He then proposed a method based on separate Single Exponential Smoothing (SES) forecasts on the size of a demand and on the time period between observing two demands, both with the same smoothing constant ?. Let ?? be the demand occurring during the time period ? and the indicator variable ?? be: ??=? 1 when demand occurs 0 when no demand occurs ? Equation ?2-1 Furthermore, let ?? be the number of periods with nonzero demand during the interval [0,?], ??=? ?? ? ?=1 . The size of the ?th non-zero demand is then shown by ?? and the inter-arrival time between ???1 and ?? is ??. Croston (1972) adopts a stochastic model of arrival and size of demand, assuming that demand sizes, ??, are normally distributed, ?(?,? 2), and that demand is random and has a Bernoulli probability 1/? of occurring in every review period (subsequently, the inter-arrival time, ??, follows the geometric distribution with a mean ?). Both demand sizes and intervals are assumed to be stationary. The demand at period ? is given by: ??=???? Equation ?2-2 Using Croston?s method, the demand size and inter-arrival time between demands M.Mohammadipour, 2009, Chapter 2 19 are separately forecasted using Single Exponential Smoothing (SES), with forecasts being updated only after demand occurrence. Let ?? and ?? be the forecasts of the (?+ 1)th demand size and inter-arrival time respectively, based on demands up to period j. Then, based on Croston?s method: ??=?1??????1 +??? Equation ?2-3 ??=?1??????1 +??? Equation ?2-4 The forecast for the next time period is then given by the smoothed size of demand divided by the smoothed inter-arrival time: ??=??/?? Equation 2-5 According to Croston (1972), the expected value and variance of the forecast for the stochastic model are given by: ?????=?/? Equation 2-6 var????= ? 2?? ? ??1 ?2 ?2 + ?2 ? ? Equation 2-7 based upon which, he claims the method is unbiased. However, Syntetos and Boylan (2001) showed that Croston?s method is biased due to the fact that ?(??)??(??)/?(??). Syntetos and Boylan (2005) suggest some modifications to reduce the bias which will be discussed in section 2.3.1.1. Rao (1973) made corrections to some of the expressions of the forecast variance. However, these changes have no effect on the forecast of mean demand. Croston?s method is based on using the same smoothing constant (?) for updating demand sizes and demand intervals. Schultz (1987) suggests using different smoothing constants, ? and ?, for size and interval between demands. M.Mohammadipour, 2009, Chapter 2 20 Croston's method is also based on assumptions of independence and normality of demand, independence of demand sizes and inter-arrival times, and independence of inter-arrival times with a Geometric distribution. The validity of these assumptions for real-world data has been discussed by several authors (e.g. Willemain et al., 1994; Snyder, 2002; Shenstone and Hyndman, 2005). Willemain et al. (1994) argue that not only might there be autocorrelation among demand sizes and inter-arrival times, but it is also possible for sizes and intervals to be correlated. Shenstone and Hyndman (2005) explore possible models underlying Croston?s method. These models are reviewed in section 2.3.1.4. 2.3.1.1 Bias Correction for Bernoulli Demand Incidence As explained in the previous section, Syntetos and Boylan (2001) showed that Croston?s forecasts are biased. A new estimator is proposed later by Syntetos and Boylan (2005) to reduce the bias associated with Croston?s method. It is shown that the bias can be approximated by ? ? 2?? ? ??1 ?2 ? as: ?? ?? ?? ?? ? ? + ? 2?? ? ??1 ?2 Equation 2-8 and based on Equation 2-8 they suggest that the forecast should be multiplied by (1? ? 2 ) in order to reduce the bias. ???1? ? 2 ? ?? ?? ?? ? ? Equation 2-9 Therefore, the new estimator of mean demand is: ??=?1? ? 2 ? ?? ?? Equation 2-10 All of Croston?s assumptions are maintained in the derivation of the new estimator M.Mohammadipour, 2009, Chapter 2 21 (however, the assumption of normality of demand sizes is not necessary for the derivation). Some of these assumptions have been changed in another study by Shale et al. (2006) which will be discussed in the next section. The new method, called the Syntetos-Boylan Approximation (SBA) in the literature, is then compared to three other forecasting methods: Simple Moving Average, Single Exponential Smoothing, and Croston?s method on 3,000 real intermittent demand data series from the automotive industry. The results suggest that the new method is the most accurate estimator for the faster intermittent demand data. Teunter and Sani (2009) also compared Croston's method, SBA and modified Croston (see section 2.3.1.3) and show that SBA has the smallest average standard deviations of all. 2.3.1.2 Bias Correction for Poisson Demand Incidence Shale et al. (2006) consider the case of Poisson demand incidence instead of a Bernoulli distribution as in Croston?s model. Also, they assume that the inter-arrival time follows a negative exponential instead of a geometric distribution. They derive the bias expected for this case and provide the correction factor for application. This is done for the cases where either a simple or an exponentially weighted moving average (EWMA) is used. Hereafter, this method is called the Shale-Boylan- Johnston (SBJ) method. For the case of a simple moving average, when the ? most recent inter-arrival times have a negative exponential distribution, the average of them has an Erlang distribution. The correction factor for an adapted Croston?s estimate of mean demand for this case is shown to be [(??1)/?]. The adaptation consists of a ratio of simple moving average of demand size and demand interval. The estimate of demand would then be: ?????=? ??1 ? ? ?(??) ?(??) Equation 2-11 M.Mohammadipour, 2009, Chapter 2 22 where ?(??) and ?(??) are the arithmetic averages rather than the exponentially weighted averages of demand sizes and interval between demands. In the latter case, it is shown that the smoothing parameter ??? of EWMA is linked to ? as follows: ?= 2?? ? Equation 2-12 Based on the above result, the estimate of demand (which will be used later in this research) is: ?????=?1? ? 2?? ? ?(??) ?(??) Equation 2-13 When ? is not an integer, the probability distribution is the continuous form of the Erlang distribution which is Gamma distribution. The correction factor for this case is also [(??1)/?]. It has been shown through simulation that these correction factors are very close to the observed average bias associated with Croston?s method (Shale et al., 2006). 2.3.1.3 Modified Croston (MC) Lev?n and Segerstedt (2004) propose a Modified Croston procedure, based on an earlier working paper by Segerstedt (2000), which they claim works for both fast- moving and slow-moving items. The procedure is given by: ?? ??=???1 ?? +?? ?? ??????1 ????1 ??? Equation 2-14 where ?? ?? is the forecasted demand rate at the end of period ??, ?? is the time period in which the demand ?? occurs, ?? is the measured demand quantity during the nth period, and ? is the smoothing constant. Demand is assumed to follow the Erlang distribution. This modified Croston procedure is then compared to Croston?s method in which: M.Mohammadipour, 2009, Chapter 2 23 ?? ?= ?? ?? Equation 2-15 ??=???1 +?(???1????1) Equation 2-16 ??=???1 +?(???1????1) Equation 2-17 Where ?? ? is the forecasted demand rate using Croston?s method and ?? is the forecasted demand interval calculated at the end of period ?? with other notations as before. The simulation results suggest that an inventory control system based on the MC procedure and the Erlang distribution yields fewer shortages than a system using exponential smoothing and the Normal distribution. The authors believe that is due to the MC procedure and not on the assumption of Erlang distribution. Lev?n and Segerstedt (2004) claim that the MC estimator avoids bias. However, Boylan and Syntetos (2007) prove that not only is it biased, but also its bias is substantially greater than the original Croston procedure, especially for highly intermittent series. The results of a comparison study by Teunter and Sani (2009) confirm the bias of the MC method, showing that it has the highest bias when compared to the original Croston?s method, SB approximation and the Syntetos approximation (Syntetos, 2001). Therefore, the MC method will not be considered further in this thesis. 2.3.1.4 Models?underlying?Croston?s?Method Shenstone and Hyndman (2005) use autoregressive integrated moving average (ARIMA) models for both size of demand and inter-arrival times. In an attempt to find an underlying model for Croston?s method, they have looked at several models. First, they assume that ??~ARIMA(0,1,1) and ??~ARIMA(0,1,1), where ?? is the size of the jth non-zero demand, ?? is the inter-arrival time between ???1 and ??, and ?? and ?? are independent. They also restrict the sample space of the underlying M.Mohammadipour, 2009, Chapter 2 24 model to be positive by defining log(??)~ARIMA(0,1,1) and log(??)~ARIMA(0,1,1). The other two models are the ones proposed by Snyder (2002) in which ??~ARIMA(0,1,1) and ??~ Geometric(?), and log(??)~ARIMA(0,1,1) and ??~ Geometric(?) where ? is the average inter-arrival time of the series. Shenstone and Hyndman (2005) suggest that any model assumed to be underlying Croston?s method must be non-stationary and defined on a continuous sample space including negative values. However, they argue that none of the above mentioned models is consistent with the properties of intermittent demand data. Finally, they suggest that instead of focusing on models based on SES, it is worth considering stationary models such as Poisson autoregressive models. These models have not been used for intermittent demand data before, opening up a new area of study. In a recent working paper, Snyder and Beaumont (2007) claim to find a logically sound model underlying Croston?s method. They assume that the sizes of positive demands follow a Poisson distribution. However, they assume that the Poisson distribution is shifted by one to the right to avoid zero demands. This results in the following probability mass function for a positive demand at period ?: ???= (???1?1) ??1 ???1?! ?????1+1 Equation 2-18 where ? is the value of the positive demand. The probability of a positive demand in period ? is then defined by ?? which is: ??= 1 ?? Equation 2-19 Finally, the probability of demand in period ? is given by: ????=??=? 1??? if ?= 0 ????? if ?> 0 ? Equation 2-20 The above equation is claimed to be the underlying model for Croston?s method i.i.d. i.i.d. M.Mohammadipour, 2009, Chapter 2 25 (Snyder and Beaumont, 2007). It can be considered as a model underlying Croston's method. However, they have not performed any assessment of the optimality of the proposed model for Croston's method. If a model exists for which Croston?s method is optimal, then it has yet to be formally established and proven. 2.3.2 Bootstrapping Bootstrapping has been proposed by Willemain et al. (2004) and Porras (2005). It is based on repeatedly sampling ? demands from demand history to estimate the distribution of lead time demand (lead time = ?). The main advantage of bootstrapping is that it forecasts the whole lead time demand distribution. There are many variants of the bootstrapping method (see for example Efron, 1979; Bookbinder and Lordahl, 1989) many of them having the disadvantage of being complex, a disadvantage that also holds for the bootstrapping method proposed by Willemain et al. (2004). The bootstrapping method proposed by Porras and Dekker (2007) is simpler. Both of these methods along with the method proposed by Snyder (2002) are discussed in this section. 2.3.2.1 Snyder (2002) Snyder (2002) proposes a parametric bootstrap method to generate an approximation for the lead-time demand distribution from Croston's model. In each iteration, first, the values for noise terms are generated from a normal distribution with mean 0 and variance ?2. Then the values for indicator variables (??, Equation 2-1) are generated from a Bernoulli distribution with probability p . A realisation of future demand series is then produced based on Croston?s method (Equation 2-3 and Equation 2-4), and finally, the lead-time demand is calculated from ? ?? ?+? ?=?+1 . In order to tackle the problem of having negative demands, Snyder (2002) suggests two adaptations. The first is to apply exponential smoothing to the logarithm of the data, which is called log-space adaptation. The other method, called the adaptive variance version, differs from Croston?s method in two respects: (i) variability is M.Mohammadipour, 2009, Chapter 2 26 measured in terms of variances instead of mean absolute deviations (MAD); (ii) a second smoothing parameter, ?, is used to define changes of variability over time. Snyder (2002) claims that the advantage of the methods is that they can be applied to both fast and slow moving demand. However, they have the disadvantages of underestimating the variability of lead-time demand due to ignoring the effects of estimation error, and being based on the Normal distribution, which may not be appropriate when demand sizes are small. 2.3.2.2 Willemain et al. (2004) Bootstrapping (Efron, 1979) produces pseudo-data by sampling with replacement from the observations. Willemain et al. (2004) develop a modified bootstrap to forecast the distribution of the sum of intermittent demands over a fixed lead time. The modification allows for autocorrelation, frequent repeated values, and relatively short series which are ignored in conventional bootstrapping. The underlying model of intermittent demand incidence that they assume is a two- state, first order Markov process. As for the demand sizes, two models have been discussed. The first model only assumes that demand sizes are stationary and they can be obtained by the method of sampling from the nonzero values that have appeared in the past. The problem here, as also mentioned by Willemain et al. (2004), is that no different values would appear in the future. As a result, they suggest an ad hoc method to deal with demand sizes called ?jittering? based on no model for demand sizes. The method is based on generating a sequence of zero and nonzero values over the L periods of the lead time. The state transition probabilities are estimated from the historical demand series using started counts. Then values are assigned to nonzero forecasts. Here, instead of only choosing nonzero values that have appeared in the past, which results in having no different values in the future, they jitter the selected value from the past, i.e. add some random variation to it. Although some methods for intermittent demand forecasting are based on the M.Mohammadipour, 2009, Chapter 2 27 assumptions that demands in each time period are independent and normally distributed, with neither of them necessarily being valid for intermittent demand (Willemain et al., 2004), Willemain et al.?s bootstrap method requires neither assumption. It only assumes that demand is stationary. The main advantage of bootstrapping is that it does not rely on any distribution while all other methods do. However, the model of Willemain et al. (2004) for demand occurrence is not general and also the method of producing variable demand sizes is ad hoc. They compare the accuracy of the developed bootstrap against exponential smoothing and Croston?s method by applying them to over 28,000 items provided by nine industrial companies. The results suggest that, although Croston?s method provides more accurate estimates of the mean level of demand at the moments when demand occur, it does not outperform exponential smoothing when forecasting the lead time distribution. However, the bootstrap provides an improvement on exponential smoothing, especially for short lead times. Gardner and Koehler (2005) argue that Willemain et al. (2004) did not use the correct lead time demand distribution for exponential smoothing and Croston's method in their comparison. They also argue that Willemain et al. (2004) should have considered the modifications to the Croston's method that have shown improvements. 2.3.2.3 Porras and Dekker (2007) Porras and Dekker (2007) propose another procedure to specify the lead time demand (LTD) distribution. Known as the Empirical Method, it differs from Willemain?s in that it constructs a histogram of demand over lead time without sampling. Therefore, it has the benefit of capturing the autocorrelation of LTD. It is also easier than Willemain?s bootstrap to implement. However, when lead time is long and the length of demand history is short, which is often the case for intermittent demand data, there are few blocks of LTD to select from. Another problem with this method is that, particularly for short lead times (say ?= M.Mohammadipour, 2009, Chapter 2 28 1), it is not possible to attain high percentiles in the demand distribution. Even if a high percentile can be reached, it might be an outlier and not representative of the population. Willemain et al. (2004) tackle this issue by introducing jittering, as explained in the previous section. An empirical study is conducted to compare the performance of the new method from an inventory control perspective to other methods including: Willemain?s bootstrap, Normal distribution, and Poisson distribution. For the normal-based model, it is assumed that demand per period follows a Normal distribution. The average and standard deviation of the observed demand are then used to estimate the parameters of the Normal lead time demand. When the lead time demand is assumed to have a Poisson distribution, the parameter is estimated from the demand data for the different items. The results show that although demand of spare parts does not generally follow a Normal distribution, the Normal model performs well, producing the highest savings in inventory costs. It has also been found that the Empirical Method outperforms the Willemain method. 2.3.3 Causal Models As pointed out by Hua et al. (2007), demand of spare parts can be attributed both to the status of the equipment and the spare parts, and to the maintenance policy. Hua et al. (2007) establish the impact of the maintenance policy through finding the relationship between explanatory variables and the nonzero demand of spare parts. They develop a method called the Integrated Forecasting Method (IFM) that first forecasts the occurrence of nonzero demand and then estimates the lead time demand. The first is done using an autocorrelation function to choose either a Markov process or explanatory variables, while the second is done by sampling from the nonzero values observed in the past and summing them over the lead time. They compare the results of the Markov Bootstrapping (MB) method (same as Willemain?s bootstrapping method except that the jittering technique is not used) with IFM. The performance of SES, Croston?s method, MB and IFM is also compared based on the mean absolute percentage error of lead time demand M.Mohammadipour, 2009, Chapter 2 29 (MAPELTD). The results confirm that Croston?s method provides more accurate estimates of mean LTD than the other methods. Ghobbar and Friend (2002) study the source of lumpiness of demand for aircraft maintenance repair parts, in order to reduce the occurrence of part shortages. They study the effect of five environmental factors on lumpiness. These factors are: the primary maintenance process (PMP) (including hard-time and conditional- monitoring), the aircraft utilization rate (AUR), the component?s overhaul life (COL), the squared coefficient of variation of demand (CV2) and the average inter- demand interval (ADI). The first three factors are independent variables and the last two are dependent variables in the general linear model (GLM). The results show that the demand variability increases when the level of aircraft utilization and flying hours increases. It shows that AUR can be a major source of lumpiness because it increases the CV2 and decreases the ADI for the observed demand. In another study, they compare the results of 13 forecasting methods including Croston?s method, SES, and also those used by aviation companies (Ghobbar and Friend, 2003). Four environmental factors considered in this study are: the seasonal period length, primary maintenance process (PMP), squared coefficient of variation (CV2) and the average inter-demand interval (ADI). CV2 and ADI are taken as covariate factors while the other two are considered as categorical factors. The variation attributable to each factor and their interactions is studied through using analysis of variance (ANOVA). Ghobbar and Friend (2003) also establish a predictive error-forecasting model (PEFM) to compare different forecasting methods based on their factor levels to evaluate which method is the best in any situation. The model is based on a GLM that predicts a response variable using its relationship with factor variables. The results of studies on causal models show an interesting line to pursue. These models are especially useful when a short demand history is available and time series methods cannot be used (Boylan and Syntetos, 2008). However, these models have not yet been well developed in the literature. As a future line of study, the effect of incorporating these models and INARMA models on the accuracy of forecasts can be investigated (see chapter 10). M.Mohammadipour, 2009, Chapter 2 30 2.3.4 Conclusions on IDF Methods Three classes of methods for intermittent demand forecasting have been reviewed in this section, namely Croston?s method and its variants, bootstrapping methods, and causal methods. Empirical studies have proved the ability of Croston?s method and variants (especially SBA) to produce reasonable forecasts (Eaves and Kingsman, 2004; Syntetos and Boylan, 2005). However, properties of these methods (such as bias) have been derived based on assumptions of independence of demand, independence of demand sizes and inter-arrival times. As mentioned, some of these assumptions are not valid for real-world data. Another problem with Croston?s method is that it has not yet been shown to be optimal for a specific demand model. Modelling intermittent demand by INARMA models makes it possible to take into account the correlation between demands. Another advantage is that optimal methods are known to exist for these models. Bootstrap methods have the advantage of having no distributional and independence assumptions. The main disadvantage of these methods is that they are rather complex. These methods along with causal methods have not been well developed in the literature and more comparative studies with the best benchmark methods (as criticized by Gardner and Koehler (2005)) need to be done to prove their performance. 2.4 Assessing Forecast Accuracy The nature of intermittent demand data makes some of the conventional accuracy measures inappropriate. For example, when one or more of the observed demands is zero, the mean absolute percentage error (MAPE) is undefined and, therefore, cannot be used. In fact, none of the relative-to-the-series accuracy measures can be used because zero observations may yield ?division by zero? problems. This includes MAPE, Median Absolute Percentage Error (MdAPE), Root Mean Square Percentage Error (RMSPE), M.Mohammadipour, 2009, Chapter 2 31 and Root Mean Square Percentage Error (RMSPE). Although the symmetric MAPE introduced by Makridakis and Hibon (1995) tackles this issue by dividing the absolute error by the average of the actual observation and the forecast: sMAPE =? ????? (??+??)/2 ? ?=1 /?? 100 Equation 2-21 it is known that it suffers from asymmetry problems in its treatment of negative and positive errors (Goodwin and Lawton, 1999). Also, if the actual value ?? is zero, sMAPE would always be 200%, which is meaningless (Syntetos, 2001). As discussed by Syntetos and Boylan (2005), all relative-to-a-base accuracy measures that relate the forecast error to a benchmark, usually na?ve 1 method, should also be discarded because the error would often be zero. The mean absolute scaled error (MASE) proposed by Hyndman and Koehler (2006) does not have the problems seen with previous measures. This measure is obtained by scaling the absolute error based on the in-sample MAE from a benchmark forecast method. Assuming the benchmark method is the na?ve method, the MASE is defined as: MASE = 1 ? ? ? ????? 1 ??1 ? ??????1? ? ?=2 ? ? ?=1 Equation 2-22 A 1MASE ? means that the forecasting method is on average better than the na?ve forecasts, and a 1MASE ? indicates that the method is on average worse than the na?ve method. As explained by Hyndman (2006), the in-sample MAE is always available and more reliably non-zero than any out-of-sample measures. The MASE would be infinite or undefined only if all historical observations are equal. However, using in-sample MAE has a disadvantage of making MASE vulnerable to outliers in the historical time series (Kolassa and Sch?tz, 2007). As a result, the MASE of two time series with the same forecasts and identical true demands during the forecast M.Mohammadipour, 2009, Chapter 2 32 horizon will be different if the two series differed in their historical demands. This makes it a more complicated metric to interpret. Syntetos and Boylan (2005) categorize the accuracy measures for the purpose of comparing methods in an intermittent demand context into two categories: absolute accuracy measures and accuracy measures relative to another method. Each of these categories is reviewed as follows. 2.4.1 Absolute Accuracy Measures These measures are calculated as a function of the forecast errors alone. Examples include the mean square error (MSE) and the mean absolute error (MAE). Theoretically, these measures can be computed for intermittent demand. However, when averaged across many time series, they do not take into account the scale differences between them (Syntetos and Boylan, 2005). Mean error (ME) can be considered as an exception to the above rule because it takes into account the sign of the error and is less susceptible to scale effects. ME is given by: ME =? ????? ? ? ?=1 Equation 2-23 If ME is divided by the average demand per unit time period, the scale dependencies are eliminated. 2.4.2 Accuracy Measures Relative to another Method These measures are calculated as a ratio to other forecasting methods. Examples include percent better (PB) and the relative geometric root mean square error (RGRMSE). The percentage better measure counts and reports the percentage of time that a given method has a smaller forecasting error than another method. M.Mohammadipour, 2009, Chapter 2 33 The RGRMSE for methods A and B in a time series is: RGRMSE = ?? (?????,?) 2? ?=1 ? 1 2? ?? (?????,?)2 ? ?=1 ? 1 2? Equation 2-24 The theoretical properties of RGRMSE are discussed by Fildes (1992). He assumes that the squared errors of a particular series have the form of: ??,?+? 2 =??,?+? 2 .??+? Equation 2-25 where (??,?+?=??+????,?+?), ??+? are assumed positive and can be thought of as errors due to the particular time period affecting all methods equally, while ??,?+? are the method?s (M) specific errors. He argues that the model of Equation 2-25 represents the case where data (and therefore errors) are contaminated by occasional outliers. He then shows that the geometric (rather than arithmetic) RMSE is independent of ??+?. It can be seen from Equation 2-24 that GRMSE is identical to GMAE because the square and the square root cancel each other (Hyndman, 2006): GRMSE = ?? (?????,?) 2? ?=1 ? 1 2 ?? (?????,?)2 ? ?=1 ? 1 2 = ? |?????,?| ? ?=1 ? |?????,?| ? ?=1 = GMAE The only issue with calculating RGRMSE by Equation 2-24 is that if ?? and ??,? are identical for a specific time period ?, the measure will be undefined. Despite this problem, RGRMSE has been suggested as an appropriate measure for intermittent demand data (see for example Syntetos and Boylan, 2005), because in most of the cases the forecasts (??,?) are not integer (although averaging methods might produce integer forecasts) and therefore, the denominator is not zero. When all of the observations are zero, this shows that there has been no demand for the item in a long time and in reality no forecasts are made for such items. Therefore, such series can be excluded from the study. M.Mohammadipour, 2009, Chapter 2 34 2.4.3 Conclusions on Accuracy Measures When comparing the performance of different forecasting methods, a range of accuracy measures should be used. This is because different measures are designed to assess different aspects. For example, MSE puts heavier penalties on higher errors while MAE is designed to lessen the effect of outliers. It has been found through forecasting competition studies such as the M-competition (Makridakis et al., 1982) and the M3-competition (Makridakis and Hibon, 2000) that the performance of different methods changes considerably depending on the accuracy measure being used. As a result, such studies have used several accuracy measures. As discussed previously, not all accuracy measures can be used for intermittent demand data. MASE has been suggested for intermittent demand studies (Hyndman, 2006). Because it is based on in-sample MAE of the na?ve method, there is no ?division by zero? problem unless all of the observations are equal. However, MASE, as well as MAE, can sometimes be misleading when comparing forecasting methods. The following example illustrates how the Zero Forecasting method (ZF) can show superiority to exponential smoothing when only MASE is concerned. The idea of using ZF as a benchmark comes from Teunter and Duncan (2009). They only used MAE and MSE, but we also include ME and MASE in our comparison. The data, the same as that presented by Hyndman (2006), are given in Table 2-2. Table ?2-2 Example data (Hyndman, 2006) In-sample Out-of-sample Actual ?? 0 2 0 1 0 1 0 0 0 0 2 0 6 3 0 0 0 0 0 7 0 0 0 0 0 0 0 3 1 0 0 1 0 1 0 0 Na?ve forecast ?? 0 2 0 1 0 1 0 0 0 0 2 0 6 3 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 We have compared Exponential Smoothing (ES) with ZF. The results of ME, MASE, MAE, and MSE (average over both in-sample and out-of-sample errors) are given in Table 2-3. Table ?2-3 Comparing ES with ZF based on different accuracy measures ME MASE MAE MSE Exponential Smoothing ?=?.? 0.1722 0.7114 1.1754 3.3458 Zero Forecast 0.8 0.4842 0.8 3.3143 M.Mohammadipour, 2009, Chapter 2 35 The results of Table 2-3 suggest that ZF is the best method when MASE, MAE and MSE are used. Teunter and Duncan (2009) suggest that therefore these accuracy measures should not be used in this regard and inventory implication metrics should be used instead. However, we believe that these measures should be interpreted based on what they are designed to measure and, as a result, different measures should be used to compare forecasting methods. MASE and MAE are designed to measure the absolute errors while ME is designed to measure bias. Because intermittent data have many zero observations, when ZF is used on these observations, the error would be zero. Therefore, ZF produces better results on MASE and MAE than ES does. But looking at ME reveals that the forecasts are much more biased than forecasts based on ES. Table 2-4 summarizes the absolute accuracy measures which will be used by this study. Table ?2-4 Accuracy measures for simulation and empirical studies Theory Simulation Empirical MSE ? ? ? ME ? ? ? MASE ? ? RGRMSE ? PB ? MSE is a widely used measure in the forecasting literature, is mathematically easy to handle, can be linked to a quadratic loss function and therefore it will be used for theoretical comparisons. MSE is a sensible measure for evaluating an individual time series and, although it is scale dependent and cannot be used for assessing a method?s accuracy across multiple series, it can be used in simulation where all the series are theoretically generated. Keeping in mind the scale-dependency and sensitivity to outliers issues, we will also use MSE for empirical analysis. ME can also be used for theoretical comparisons and, as discussed earlier, it does not have the scale-dependency problem. Therefore it will be used for both simulation and empirical analysis. Based on the above mentioned properties, MASE will be used for simulation and empirical analysis. It should be mentioned that as both MAE and MASE are used to M.Mohammadipour, 2009, Chapter 2 36 measure the same factor (absolute errors), we do not see any benefit in using both and only MASE will be calculated. In addition to the above measures, two relative-to-another-method measures will be used for empirical analysis: PB and RGRMSE. The PB is based on comparing the MASEs of two methods and determines how often a method is better than another, but not by how much. RGRMSE is used to calculate the magnitude of improvement (in terms of MSE) of one method over another. Both of these measures have been used in an intermittent demand context (Syntetos and Boylan, 2005; Teunter and Duncan, 2009). They are insensitive to outliers and have been used in other studies, allowing comparative analyses to be undertaken. 2.5 Comparative Studies A number of studies have been undertaken to compare methods for forecasting intermittent demand (Willemain et al., 1994; Johnston and Boylan, 1996; Eaves and Kingsman, 2004; Lev?n and Segerstedt, 2004; Willemain et al., 2004; Syntetos and Boylan, 2005). Three of the main comparison studies will be discussed here. Eaves and Kingsman (2004) compare the performance of exponential smoothing (ES), Croston?s method, Syntetos-Boylan Approximation (SBA), a moving average (MA12) and a simple average method. Mean absolute deviation (MAD), root mean square error (RMSE) and MAPE have been used to compare the accuracy of forecasts obtained by each method, although it has been concluded that these measures are not ideal for slow-moving demands. It is also suggested to use stock- holdings as a measure instead of the above-mentioned conventional accuracy measures. The results show the superiority of SBA with regards to stock-holdings. The study by Willemain et al. (2004) compares three methods of ES, Croston?s, and Willemain?s bootstrap based on the uniformity of observed lead-time demand (LTD) percentiles. It is found that Croston?s method provides a more accurate estimate of mean demand than ES, but the same is not true for forecasting the distribution of LTD. The results also show that bootstrapping is the most accurate method, especially for short lead times. M.Mohammadipour, 2009, Chapter 2 37 Another study by Syntetos and Boylan (2005) compares the accuracy of forecasts obtained by simple moving average (SMA13), SES, Croston?s method, and SBA. ME, scaled mean error, RGRMSE, Percentage Better (PB), and Percentage Best (PBt) have been used and the results suggest that using different accuracy measures leads to different conclusions, agreeing with the findings of Eaves and Kingsman (2004). However, the SBA seems to be the most accurate method for faster intermittent demand (p values close to 1). It is also argued by Teunter and Duncan (2009) that the conflicting results of comparative studies in the literature are due to use of inappropriate measures. Instead of comparing ?per period forecast error? (measures such as MAD and MSE), they propose analyzing the effect of forecasting methods on inventory control parameters and also comparing the resulting average inventory and service levels. Boylan and Syntetos (2006) also highlight the distinction between these two approaches which they refer to as forecast-accuracy metrics and accuracy-implication metrics, respectively. As discussed previously, because this research only focuses on forecasting and not on inventory control, the forecast-accuracy metrics suggested in section 2.4 will be used. The proposed method based on INARMA modeling of intermittent demand will be compared with Croston?s method, SBA and SBJ. The reason for considering Croston?s is its simplicity, popularity (for example, it is used in Forecast Pro), and its superiority over SES and SMA (Syntetos and Boylan, 2005). SBA has also been proved by a number of studies discussed above to perform reasonably well and therefore is included for comparison in this research. SBJ is included because it is based on Poisson demand arrivals. The reason for ignoring bootstrapping is that we have restricted our model to Poisson marginal distribution which makes it an inappropriate comparison considering the fact that there is no distributional assumption for bootstrapping. 2.6 INARMA Models As discussed by Willemain et al. (1994), stochastic models of intermittent demand have assumed that the successive intervals between demands, successive demand M.Mohammadipour, 2009, Chapter 2 38 sizes, and intervals and sizes are mutually independent. However, their sample data contradict this assumption (with some autocorrelations as high as -0.53 and 0.39). This emphasizes the necessity to use models that take into account autocorrelation. Shenstone and Hyndman (2005) also suggest using integer autoregressive moving average (INARMA) models with Poisson marginal distribution or other time series models for counts for intermittent demand forecasting. This research focuses on using a class of models called INARMA models to model intermittent demand. The forecasting method will then be the minimum mean square error (MMSE) of the model. We start from a stochastic model for intermittent demand and then build a method for forecasting which is optimal for the underlying model. It has also the advantage of enabling us to directly find the mean and variance of lead-time demand. The conditional mean of lead-time demand will be compared with the lead-time forecasts of the benchmark methods in chapters 8 and 9. Table 2-5 summarizes the methods for intermittent demand forecasting based on their dependence, distributional, and stationarity assumptions. Although this research only focuses on INARMA models with Poisson marginal distribution, other discrete distributions such as negative binomial can be considered as a future line of study. Table ?2-5 The categorization of methods of intermittent demand forecasting based on their assumptions Models Dependence structure Distribution Croston?s method independent Normal sizes of demand Geometric inter-arrival times SBA independent Normal sizes of demand Geometric inter-arrival times SBJ independent Geometric sizes of demand Negative Exponential or Erlang inter-arrival times Bootstrapping: Willemain et al. (2004) Markov demand incidence - Porras and Dekker (2007) - INARMA modelling autocorrelated demand Poisson 2.7 Conclusions The literature on forecasting intermittent demand has been reviewed in this chapter. Different definitions of intermittent demand have been reviewed. In this study, a M.Mohammadipour, 2009, Chapter 2 39 series is called intermittent if it contains non-negative integer values with some values being zero (Shenstone and Hyndman, 2005). As explained, intermittence and lumpiness are two different concepts and should be distinguished. This study focuses on INARMA models with a Poisson marginal distribution. Although other marginal distributions could be used to allow for more demand size variability, in general these models are not designed for very lumpy demand. The two classification schemes for intermittent demand by Eaves and Kingsman (2004) and Syntetos et al. (2005) have been reviewed. As explained, the INARMA approach is universal and can be used for both intermittent and smooth demand but we will use the Syntetos et al. classification to find the best benchmark method for comparison. Different intermittent demand forecasting methods have been reviewed. These methods are: ? Croston?s method and its variants ? Bootstrapping ? Causal methods Although Croston's method is the most widely used approach for intermittent demand forecasting, it has been shown that it is biased (Syntetos and Boylan, 2001). Some modifications have been suggested to reduce its bias (e.g. SBA and SBJ) while some add even more bias (Modified Croston). The optimal demand model underlying Croston's method has not yet been established, although some studies have focused on finding such a model (Shenstone and Hyndman, 2005; Snyder and Beamont, 2007). Different bootstrapping methods have been established in the literature to estimate the distribution of lead time demand. The parametric bootstrap by Snyder (2002) has the disadvantage of being based on the Normal distribution. The method of Willemain et al. (2004), however, is not based on any distributional assumptions. Porras and Dekker (2007) develop another procedure for identification of the lead time demand. The studies on using causal models for intermittent demand forecasting have also been reviewed. Both bootstrapping methods and causal models seems promising but the literature on these methods is not yet well-developed. As will be discussed in chapter 10, incorporating causal factors into INARMA models can be pursued as a M.Mohammadipour, 2009, Chapter 2 40 future line of study. The performance of INARMA models in forecasting the lead time distribution can also be compared to that of bootstrapping methods. The fact that intermittent data include zero values makes some of the conventional accuracy measures inappropriate. Different accuracy measures that can be used for intermittent data are reviewed. Three absolute accuracy measures will be used in this research: ME, MSE, and MASE. For the empirical study, two relative-to-another- method accuracy measures will be added to the above-mentioned measures: PB and RGRMSE. Using the INARMA procedure for intermittent demand forecasting has the advantage of utilizing a model-based forecasting method which takes into account the correlation between demands. Unlike bootstrapping, a marginal distribution has to be assumed. The Poisson distribution has been selected because it has been used in the IDF literature and also due to its interesting properties in INARMA models, to be reviewed in chapter 3. As discussed in section 2.5, the results of INARMA method will be compared to Croston?s method, Syntetos-Boylan Approximation (SBA) and Shale-Boylan- Johnston (SBJ) method. The next chapter introduces INARMA models and reviews their properties in detail. M.Mohammadipour, 2009, Chapter 3 41 Chapter 3 INTEGER AUTOREGRESSIVE MOVING AVERAGE MODELS 3.1 Introduction Continuous-valued autoregressive integrated moving average (ARIMA) models, also known as Box-Jenkins models, developed by Box and Jenkins (1970) are used to model stationary processes under the assumption of Gaussianity, i.e. all the joint distributions of the time series are multivariate normal (Brockwell and Davis, 1996). However, this assumption is not valid for modelling count data, especially for low- frequency count data that cannot be suitably approximated by continuous models. Therefore, a number of models using different approaches have been proposed for integer-valued time series in the literature (e.g. Smith, 1979; Zeger, 1988; Zeger and Qaqish, 1988; Harvey and Fernandes, 1989). M.Mohammadipour, 2009, Chapter 3 42 These models are divided into two categories: observation-driven and parameter- driven. Observation-driven models specify a direct link between current and past observations, while parameter-driven models rely on a latent process connecting the observations (Jung and Tremayne, 2006a). As an example of parameter-driven models for time series of counts, Zeger and Qaqish (1988) proposed Poisson regression to model trend and seasonality explicitly and an unobserved stationary process (latent process) to model the autocorrelation. One of the early observation-driven models for count data is the model suggested by Jacobs and Lewis (1978a; 1978b; 1983) called DARMA (discrete autoregressive moving average) models which will be discussed in section 3.2. Another class of observation-driven models has been developed by McKenzie (1985) and later generalized by Al-Osh and Alzaid (1987) as integer-valued autoregressive (INAR) models for modelling series with correlated counting data. Examples of this kind of time series include the number of patients in a hospital at a specific point of time, the number of people in a queue waiting to receive a service at a certain moment (Silva and Oliveira, 2004), and the number of accidents in a firm each three months (McKenzie, 2003). Table 3-1 briefly describes the most frequently used count models and summarizes their strengths and weaknesses (McKenzie, 2003; Rengifo, 2005). This thesis is focused on integer autoregressive moving average (INARMA) models. As explained in chapter 2, the need for a model-based method for intermittent demand forecasting has motivated this research. The similarities that these models have with the conventional ARMA models are also an advantage. Because DARMA models are also based on ARMA models, we introduce them in section 3.2. The main reasons for excluding these models from our study are explained in this section. This chapter mainly focuses on introducing the INARMA models and reviewing their statistical properties. Identification of these models, estimation of their parameters and INARMA forecasting will be discussed in subsequent chapters. This chapter is structured as follows. In section 3.2, DARMA models are introduced. M.Mohammadipour, 2009, Chapter 3 43 INARMA models are reviewed in section 3.3. Summary of literature review and the conclusions are provided in sections 3.4 and 3.5. Table ?3-1 Review of count models in time series Model Description Strengths and weaknesses Markov chains First, the transition probabilities between all the possible values that the count variable can take are defined. Then, the appropriate order of the time series is determined (e.g. see Raftery 1985, and Pegram 1980). This method can only be reasonably used when the possible values that the observations can take are very limited. It is generally overparametrized and too limited in correlation structure. Discrete Autoregressive Moving Average (DARMA) models These models are structurally based of ARMA models and are probabilistic mixtures of discrete i.i.d. random variables with a marginal distribution (see section 3.2). When the serial correlation is high, the data will be characterized by a series of runs of a single value. Therefore, they are rarely used. The main application is in the hydrological literature. Integer Autoregressive Moving Average (INARMA) models These models are a generalization of the linear ARMA models for count data (see section 3.3). The models have the same serial correlation structure as ARMA models. The marginal distribution of the model is same as the distribution of the innovations if the latter is Poisson. Regression models (or generalized linear models) These are regression models for the special case where the dependent variable is a non- negative integer with a correction for autocorrelation (Zeger, 1988; Br?nn?s and Johansson, 1994). These models are generally easy to construct and have the ovserdispersion property. An explicit joint density function cannot be obtained which restricts the class of possible predictors. The hidden Markov models These models are extension of the basic Markov chains models, in which various regimes characterizing the possible values of the mean are identified. It is then assumed that the transition from one regime to another is ruled by a Markov chain (MacDonald and Zucchini 1997). There is no accepted way of determining the appropriate order for the Markov chain. There can be too many parameters to estimate, especially when the number of regimes is large. State Space models These models specify the conditional distribution of the series to depend on stochastic parameters that evolve according to a specified distribution. The parameters of such distributions are determined by some regressors (Harvey and Fernandes, 1989; Durbin and Koopman, 2000). These models are very general and can be used in a wide range of applications. The behaviour of different components of the series can be modelled separately and then put together. 3.2 DARMA Models This class of models was first introduced by Jacobs and Lewis (1978a) for a stationary sequence of dependent discrete random variables. Because we focus on INARMA models in this thesis, only a brief review of the first order discrete autoregressive model is provided in this section. The applications of DARMA models will then be discussed. M.Mohammadipour, 2009, Chapter 3 44 3.2.1 DAR(1) Model The first order discrete autoregressive form, DAR(1), is given by: ??=?????1 + (1???)?? Equation ?3-1 where {??} are i.i.d. binary random variables with ????= 1?=? and {??} are i.i.d. discrete random variables with the probability mass function (hereafter called a ?distribution? as by the authors) ?1 (Jacobs and Lewis, 1978a). Therefore, in this model, the current observation is either the previous observation, with probability ?, or another, independent, sample from a specific distribution. This approach is similar to the Box-Jenkins approach except that a probabilistic mixture replaces the linear combination in the continuous-valued model and, as the authors mentioned, a realization of the process will generally contain many runs of a constant value. This can be a significant disadvantage of these models. It is obvious that the larger the value of ?, the longer the runs. It can be seen that, in the model of Equation 3-1, ???1 contains all information about the past. Because there is randomization between ???1 and ??, if ?? is chosen, the memory of the process before time ? is gone forever. The autocorrelation function (ACF) of {??} is given by: ??=? ? for ?= 0,1,? Equation ?3-2 The conditional mean of ?? given ???1 is linear in ???1 and the conditional variance is quadratic in ???1. It also follows from Equation 3-1 that {??} is a Markov chain since: ????+1 =??1,?,???=????+1 =???? Equation 3-3 The transition matrix for this Markov chain is given by: 1 ????=??=?(?) ?= 0,1,? M.Mohammadipour, 2009, Chapter 3 45 ????+1 =???=??=? ?1????(?) for ??? ?+?1????(?) for ?=? ? Equation 3-4 It can be seen from Equation 3-1 that when ?= 0, the model is reduced to a stationary sequence of i.i.d. random variables with distribution ?. The mixed DARMA(p,q+1) model is built by adding the two autoregressive and moving average components as follows: ??=???????+ (1???)?????1 Equation 3-5 ??=???????+ (1???)?? Equation 3-6 where {??} and {??} are as before. {??} are i.i.d. random variables defined on the set ?= {1,2,?,?} in such a way that ????=??=??. {??} are i.i.d. random variables so that ????=??=?? for ?= {0,1,?,?} and {??} are i.i.d. binary random variables with ????= 1?=?. 3.2.2 Applications of DARMA Models Applications of DARMA models can be found mainly in the hydrological literature. This is not surprising given the structure of the model, which represents dependence as runs (McKenzie, 2003). These applications are briefly discussed below. In climatic analysis, these models were introduced by Buishand (1978). He proposes a binary discrete autoregressive moving average (B-DARMA) process to model daily rainfall sequences and finds that this model is promising in tropical and monsoonal areas. Chang et al. (1984) use the same type of models (B-DARMA) in their study of daily precipitation by transforming the daily level of precipitation into a discrete variable based on its magnitude. They conclude that the statistical properties of the daily rainfall process can be preserved by DARMA models. M.Mohammadipour, 2009, Chapter 3 46 Salas et al. (2001) use DARMA models to simulate the return period and risk of extreme droughts. A series of wet and dry years is obtained from a continuous-valued hydrologic series, such as annual stream flows, and a method is presented for relating the autocorrelation functions of these two series. The analysis of 23 series of annual flows reveals that this relationship is applicable and reliable. Ksenija (2006) employs a DARMA(1,1) model to describe the wet-dry day sequences in Split, on the middle Adriatic coast of Croatia. The results are compared to the DAR(1) model and reveal that, although both models underestimate dry spell runs, the DARMA(1,1) model provides a better fit to the empirical distribution both for short (one day) and long runs (more than 10 days). But, for short wet spells, the DAR(1) model estimates are closer to the observed frequencies of short spells in the months studied. As previously mentioned, the main disadvantage of the DARMA models, which makes their application areas very limited, is that a realization of the process will generally contain many runs of a constant value. This is especially true when the serial correlation is high. Therefore we exclude these models and concentrate on another class of models for count data, called integer autoregressive moving average models, which overcome these problems. 3.3 INARMA Models The integer-valued autoregressive models are equivalent to autoregressive models for Gaussian time series. By ?equivalent? we mean that they share some similar properties, which will be discussed later in relevant subsections. First, we introduce the first-order integer autoregressive, INAR(1), model. It has been shown that this model belongs to a more general class of models known as non- Gaussian conditional linear AR(1), CLAR(1), models (Grunwald et al., 2000). Other integer autoregressive, moving average and mixed models are then described. Although many papers discuss the statistical aspects of INARMA models, fewer studies have been done regarding their practical application (Jung and Tremayne, M.Mohammadipour, 2009, Chapter 3 47 2003). Some applications of these models are reviewed at the end of this section. 3.3.1 INAR(1) Model Before describing the INAR(1) model, we first introduce the meaning of the binomial thinning operation defined by Sueutel and van Harn (1979). Suppose Y is a non-negative integer-valued random variable. Then, for any ??[0,1], the thinning operation ??? is defined by: ???=? ?? ? ?=1 Equation 3-7 where {??} is a sequence of i.i.d. Bernoulli random variables, independent of ?, and with a constant probability that the variable will take the value of unity: ????= 1?= 1?????= 0?=? Equation 3-8 From the above definition, some of the properties of the thinning operation can be obtained as follows: (1) 0??= 0 (2) 1??=? (3) ??(???) (??)?? (4) ??????=??(?) (5) ??????2 =?2???2?+??1????(?) (6) var?????=?2var???+??1????(?) where stands for equal in distribution. Now, the integer-valued first order autoregressive, INAR(1), model is defined by the Equation 3-9. A discrete time stochastic process, ????, is called an INAR(1) process if it satisfies the equation: ??=?????1 +?? Equation 3-9 d = d = M.Mohammadipour, 2009, Chapter 3 48 where ??[0,1] and {??} is a sequence of i.i.d. non-negative integer-valued random variables, independent of ?? with mean ?? and finite variance ?? 2. ?? and ???1 are assumed to be stochastically independent for all points in time. The process obtained by the Equation 3-9 is stationary and it resembles the Gaussian AR(1) process except that it is nonlinear due to the thinning operation replacing the scalar multiplication in continuous models. It should be noted that, in Equation 3-9, subsequent thinning operations are performed independently of each other. Equation 3-9 shows that, based on the definition of the thinning operation, unlike the DAR(1) model, the memory of an INAR(1) model decays exponentially (Al-Osh and Alzaid, 1987). The two independence limitations we have assumed so far ? independence of {??} in the thinning operation, and independence of ?? and ???1? have been relaxed in a study by Br?nn?s and Hellstr?m (2001). It is worth mentioning that the probability ? is assumed to be constant here. Alzaid and Al-Osh (1993) develop a model in which this probability of retaining an element is not constant. Also, Zheng et al. (2007) develop a random coefficient model where ?? are i.i.d. random variables that can take values in the interval [0,1). A realization of ?? in an INAR(1) model of Equation 3-9 has two components: (i) the survivors of elements of the process at time (??1), ???1, each with probability of survival ? and (ii) the innovation term, ??, which represents new entrants to the system in the interval (??1,?]. The mean and variance of the process {??} are: ?????=? ??(?0) +??? ? ? ??1 ?=0 Equation 3-10 var????=? 2?var(?0) +?1???? ? 2??1 ? ?=1 ???????+?? 2? ?2(??1) ? ?=1 Equation 3-11 It is shown by Al-Osh and Alzaid (1987) that the autocorrelation function (ACF) of M.Mohammadipour, 2009, Chapter 3 49 this process is given by ??=? ? for ?= 1,2,?. This is identical to the ACF of a linear Gaussian AR(1) process. The only difference is that the ACF of the INAR(1) model is always positive. The derivation of the first and second order unconditional moments of the INAR(1) process is straightforward: ?????=??/(1??) Equation 3-12 var????= (???+?? 2)/(1??2) Equation 3-13 It can be seen that both the first order (regression function) and the second order conditional moments are linear in ???1. ????|???1?=????1 +?? Equation 3-14 var???|???1?=?(1??)???1 +?? 2 Equation 3-15 Another similarity between the model of Equation 3-9 and the Gaussian AR(1) is that the distribution of the innovation term (??) plays the same role in determining the distribution of ?? that the normal distribution of the shock term plays in the AR(1) model. In fact, Al-Osh and Alzaid (1987) argue that the distribution of ?? is uniquely determined by the distribution of ??. In Equation 3-9, {??} can have any non-negative discrete distribution. A natural first choice of interest for these variables is Poisson. Al-Osh and Alzaid (1987) show that if ??~??(?), the marginal distribution of the process ?? is also Poisson ??~??(?/(1??)). In this case the model is called PoINAR(1) (Jung and Tremayne, 2006b) (or PAR(1) as suggested, for example, by Freeland and McCabe (2004b)). Hence, the role of the Poisson distribution in the INAR(1) process is analogous to the role of the Gaussian distribution in the AR(1) process. The INAR(1) process is a member of a class of models introduced by Grunwald et al. (2000). They suggest that nearly all the non-Gaussian AR(1) models are in fact a part of a class of conditional linear AR(1) models, CLAR(1). M.Mohammadipour, 2009, Chapter 3 50 If ????,?= 0,1,? is a time-homogeneous first-order Markov process on a sample space ???, then it is said to have CLAR(1) structure if it satisfies: ?????1?=????1 +? where ?????1?=?(??|???1) and ? and ? are real numbers. Grunwald et al. (2000) show that stochastic properties of CLAR(1) models are similar to those of the Gaussian AR(1) model. 3.3.2 INAR(2) Model To take into account higher order dependence in the data, higher order INAR models are developed. In this section, the second order model, INAR(2), will be reviewed briefly. A discrete time stochastic process, {??} is called an INAR(2) process if it satisfies the equation: ??=?1????1 +?2????2 +?? Equation 3-16 with all the previously mentioned definitions, except for the thinning mechanism. There are two approaches regarding the binomial thinning mechanism. The first approach is proposed by Alzaid and Al-Osh (1990) and the other by Du and Li (1991). The corresponding processes will henceforth be denoted by INAR(2)-AA and INAR(2)-DL, respectively, as in Jung and Tremayne (2006b). In the INAR(2)-AA process, the random variables ?1????2 and ?2????2 which are elements of ???1 and ??, are connected in a powerful way. In fact, ??1?? and ??2?? and ???2 are dependent although they appear in different times. The vector (?1? ??,?2???) given ??=?? is multinomial with parameters (?1,?2,??). To understand this structure, consider the simulated process: at time ?, ?? is observed and from the previous time period we have ???1 and we have formed ?1????1, and ?2????1. Now, we form ??=?1??? and ??=?2???. Then we have ??+1 =??+???1 + ??+1 and ?? is available to obtain ??+2. M.Mohammadipour, 2009, Chapter 3 51 This dependence results in two important consequences: (i) the process maintains its physical interpretation in terms of the counts evolving as a birth and death process, and (ii) a moving average structure is included into the process in such a way that the ACF of the INAR(2)-AA process mimics that of a Gaussian ARMA(2,1) process (Alzaid and Al-Osh, 1990). The stationarity condition for the INAR(2)-AA process is ?1 +?2 < 1 and, assuming ??~??(?), the marginal distribution of the process ?? is ??~??(?/(1? ?1??2)). However, the conditional mean function for this model is not linear, which shows that this process is not a member of the CLAR class. The ACF of the INAR(2)-AA process is given by: ??=?1???1 +?2???2 for ??2 Equation 3-17 where the starting values are ?0 = 1 and ?1 =?1 (Jung and Tremayne, 2006b). The other specification of the INAR(2) model is that of Du and Li (1991). The concept of their model is closer to the Gaussian higher order AR models in which ?? is obtained by a direct multiplication of the constants ?1 and ?2 to ???1 and ???2, independent of all previous stochastic structures. It means that at time ? we have already observed ???1 and ???2 and the thinning operations are applied independently of the previous period which result in: ??= (?1????1) + (?2????2) +??. Du and Li (1991) show that the unconditional mean of the process is again ??/(1??1? ?2) and the stationarity condition remains the same. However, they show that the correlation properties of their model are identical to those of the Gaussian AR(2) model which is one of the main differences that distinguishes INAR(2)-DL from INAR(2)-AA. An additional difference is that the conditional mean of the ?? in INAR(2)-DL is given by: ?(?????1,???2,??=?1???1 +?2???2 +?? Equation 3-18 which obviously is linear, while that of the INAR(2)-AA process is nonlinear. Therefore, the INAR(2)-DL model has a CLAR(2) structure. It should also be noted that, even with Poisson innovations, the marginal distribution of ?? is not Poisson, M.Mohammadipour, 2009, Chapter 3 52 which is in contrast with the INAR(2)-AA model. Because the specification by Du and Li (1991) is similar to the conventional AR(2) model and we do not use the physical interpretation of the INAR models (as a birth and death process which is maintained in the specification by Alzaid and Al-Osh (1990)), we use the Du and Li approach in this study. 3.3.3 INAR(p) Model Realizations of some counting process {??} might be attributed not only to its immediate predecessors {???1} and {???2} as in INAR(2), but also to previous realizations of the process, {????}?=3 ? . The pth order integer-valued autoregressive, INAR(p), process is defined by Alzaid and Al-Osh (1990) as follows: ??=?1????1 +?2????2 +?+???????+?? Equation 3-19 with all the previously mentioned definitions. {??} are non-negative constants such that the process remains stationary and ?1,?,???1 ?[0,1] and ???(0,1]. The stationarity condition for the INAR(p) process is that the roots of the equation ????1? ??1??????1????= 0 lie inside the unit circle (Alzaid and Al-Osh, 1990). Like the INAR(2), there are two approaches concerning the binomial thinning mechanisms. The first model is that of Alzaid and Al-Osh (1990) which will be denoted by INAR(p)-AA and the other is the model of Du and Li (1991), known as INAR(p)-DL. The idea behind these approaches is the same as that explained for INAR(2). The INAR(p)-AA process shares the same correlation properties with the Gaussian ARMA(p,p-1) process (Alzaid and Al-Osh, 1990), while the INAR(p)-DL mimics the AR(p) process (Du and Li, 1991). The unconditional first moment of the INAR(p)-AA process is given by: ?????=? ??????? ? ?=1 +?? Equation 3-20 M.Mohammadipour, 2009, Chapter 3 53 The conditional moments of ?? are given by: ????|???1,?,?????=? ????? ? ?=1 +?? var???|???1,?,?????=? ??(1???)???? ? ?=1 +?? 2 Assuming ??~??(?), the marginal distribution of the process ?? would be ??~??(?/(1-? ?? ? ?=1 )). The autocovariance at lag ? of the INAR(p)-AA process satisfies the equation: ??=? ????? ? ?=1 + ? ????,?? ? ?=?+1 +??(0)?? 2 Equation 3-21 where ????,???cov?????+?,?????????? and ???0?= 1 if ?= 0 and zero otherwise. The autocovariance of the INAR(p)-AA has the same form of that of the Gaussian ARMA(p,p-1). As for the INAR(2)-DL, when the approach of Du and Li (1991) is taken, the marginal distribution of ?? is not the same as the distribution of innovations. The autocovariance function of the INAR(p)-DL satisfies: ??=?1???1 +?2???2 +?+?????? Equation 3-22 Therefore, the ACF of this process is found from equations of the form: ??=?1???1 +?2???2 +?+?????? Equation 3-23 which implies that the correlation structures of INAR(p) and AR(p) processes are the same. This makes these models similar to the standard AR(p) models not only in form, but also in stationarity conditions and correlation structure. M.Mohammadipour, 2009, Chapter 3 54 3.3.4 INMA(1) Model Having introduced integer autoregressive models for count series, Al-Osh and Alzaid (1988) then developed a class of models for integer-valued moving average (INMA) processes. In INMA models, a stationary sequence of random variables {??} is formed from a sequence {??} of i.i.d. random variables which are non-negative and also integer-valued. The first order model, which we are going to describe in this section, is the case in which adjacent members of the sequence are correlated. A process {??} is called an INMA(1) process if it satisfies the equation: ??=?????1 +?? Equation 3-24 where ??[0,1] and {??} are as before and the thinning operation is defined via: ???=? ?? Z ?=1 Equation 3-25 where {??} is a sequence of i.i.d. Bernoulli random variables, independent of ? and satisfying: ????= 1?= 1??(??= 0) =? Equation 3-26 The INMA(1) model defined by Equation 3-24 is similar to the Gaussian MA(1) process except that scalar multiplication is replaced by the thinning operation. Jung and Tremayne (2006a) present a physical interpretation of this model as follows. If we consider ?? as the number of particles in a well-defined space at time point ?, it can be assumed that this number is made of two components: (i) particles entering during (??1,?], and (ii) survivors of those who entered the space during (??2,?? 1]. Therefore, the thinning at time ?, is applied to only immigrants at time ??1, not all particles in space, as in an INAR(1) process. Examples of this process include the number of patients staying in a hospital or the number of customers in a department store (Al-Osh and Alzaid, 1988). It can be inferred from the Equation 3-24 that each element stays in the system no longer than two periods. This is in contrast to the INAR(1) process in which there is M.Mohammadipour, 2009, Chapter 3 55 no limit on the survival of elements in the system. The unconditional first and second moments of the INMA(1) process are: ?????= (1 +?)?? Equation 3-27 var????=??1?????+ (1 +? 2)?? 2 Equation 3-28 It is shown by Al-Osh and Alzaid (1988) that the autocorrelation function (ACF) of this process is given by: ?? INMA (1) =? ??? 2 ??1?????+ (1 +?2)?? 2 for ?= 1 0 for ?> 1 ? Equation 3-29 which is analogous (but not identical) to that of the Gaussian MA(1) process, where ??=????1 +?? and the ACF is given by: ?? MA (1) =? 1 for ?= 0 ? 1 +?2 for ?= ?1 0 for ??> 1 ? Equation 3-30 Another property of the INMA(1) process which is similar to MA(1) is that if ?= 0, the sequence {??} becomes a sequence of i.i.d. random variables with the distribution of ??. Also, if ?= 1 the process will have the highest ?1 which again agrees with the MA(1) process. As for the INAR(1) process, a natural candidate for the marginal distribution of an INMA(1) process is the Poisson distribution. It is shown by Al-Osh and Alzaid (1988) that assuming ??~??(?), the marginal distribution of the process ?? would be ??~??(?(1 +?)). This process is referred to as a PoINMA(1) process (Jung and Tremayne, 2006a). M.Mohammadipour, 2009, Chapter 3 56 3.3.5 INMA(2) Model The second-order moving average process is an extension of the INMA(1) process introduced in the previous section. The model is given by: ??=?1????1 +?2????2 +?? Equation 3-31 where both parameters ?1 and ?2 lie in the interval [0,1]. The individual thinning operations ??????? for ?= 1,2 follow the Equation 3-25 and it is assumed that they perform independently of each other. Two approaches arise regarding the thinning mechanisms, as in higher order autoregressive models. One approach is proposed by Al-Osh and Alzaid (1988), assuming dependence between the thinnings of terms ?1??? and ?2??? (similar to that of INAR(2)-AA). The unconditional expected value and variance of such a process, henceforth called INMA(2)-AA, are given by: ?(??) =??(1 +?1 +?2) Equation 3-32 var????=??? ???1???? 2 ?=1 +?? 2? ?? 2 2 ?=0 Equation 3-33 with ?0 = 1. The ACF of the INMA(2)-AA process is: ??=? ? [??(?????+?)?? 2?? ?=0 +????+??? 2] ??? ??(1???) 2 ?=1 +?? 2? ?? 22 ?=0 for ?= 1,2 0 for ?> 2 ? Equation 3-34 It can be seen that the cut-off property of the INMA(2)-AA process is the same as that of the Gaussian MA(2) process. The other approach concerning the thinning operation in an INMA(2) process is introduced by McKenzie (1988). In an INMA(2)-MK process, it is assumed that the individual thinning operations ??????? for ?= 1,2 are performed independently not only from each other, but also from corresponding operations at previous times in M.Mohammadipour, 2009, Chapter 3 57 Equation 3-31. The unconditional moments of this process with Poisson innovations are the same as INMA(2)-AA, ?????= var????=?(1 +?1 +?2), and the ACF, which is again the same as that of the Gaussian MA(2) process, is given by: ??=? ? ????+? 2?? ?=0 1 +?1 +?2 for ?= 1,2 0 for ?> 2 ? Equation 3-35 Unlike INAR(2)-DL, for an INMA(2)-MK process, if ??~??(?), then ??~??(?(1 +?1 +?2)) which is the same as INMA(2)-AA. In this PhD thesis, we adopt the approach by McKenzie (1988) because his model is more similar to the classic MA(q) model (Br?nn?s and Hall, 2001). 3.3.6 INMA(q) Model The ?th order integer moving average model, introduced by Al-Osh and Alzaid (1988) and McKenzie (1988) is defined by: ??=?1????1 +?2????2 +?+???????+?? Equation 3-36 where {??} is defined as before and the parameters ?1,?,???1 ?[0,1] and ???(0,1]. Using the properties of the thinning operation, it is shown by Br?nn?s and Hall (2001) that: ?(??) =??(1 +? ?? ? ?=1 ) Equation 3-37 var????=?? 2 +? [?? 2?? 2 +????(1???)] ? ?=1 Equation 3-38 As for an INMA(2) process, there are two approaches based on the thinning mechanisms. The ACF of the INMA(q)-AA process is given by: M.Mohammadipour, 2009, Chapter 3 58 ??=? ? [??(?????+?)?? ??? ?=0 +????+??? 2] ??? ??(1???) ? ?=1 +?? 2? ?? 2? ?=0 for ?= 1,?,? 0 for ?>? ? Equation 3-39 On the other hand, the ACF of the INMA(q)-MK process is: ??=? ? ????+? ??? ?=0 ? ?? ? ?=0 for ?= 1,?,? 0 for ?>? ? Equation 3-40 It can be seen that the autocorrelation function of an INMA(q) process is analogous to than of the classical MA(q) process. The difference is that all autocorrelations are positive (Br?nn?s and Hall, 2001). 3.3.7 INARMA(1,1) Model Having introduced INAR and INMA processes, Alzaid and Al-Osh (1990) suggested that these two processes can be mixed in a manner similar to that of the standard ARMA processes to provide the mixed integer autoregressive moving average class of models. There are two approaches regarding the modelling of this kind of processes. The first approach was introduced by McKenzie (1988) for INARMA processes with Poisson marginal distributions. He suggests the mixed process should be constructed by coupling the two AR and MA processes and a common innovation process. According to this viewpoint, the AR component of the INARMA(1,1) process is given by: ??=?????1 +?? Equation 3-41 and the MA component is: ??=???1 +???? Equation 3-42 M.Mohammadipour, 2009, Chapter 3 59 where all the thinning operations are independent, ?,??[0,1] and {??} is a sequence of i.i.d. Poisson variables. The second approach suggested by Neal and Rao (2007) is what we follow because it is similar to the Gaussian ARMA process. A discrete time stochastic process, {??}, is called an INARMA(1,1) process if it satisfies the equation: ??=?????1 +??+?????1 Equation ?3-43 where ?,??[0,1] and {??} is a sequence of i.i.d. non-negative integer-valued random variables, independent of ?? with mean ?? and finite variance ?? 2. Here, the two thinning operations are independent of each other and also of the corresponding operations at previous times, and are defined as follows: ???=? ?? ? ?=1 Equation 3-44 ???=? ?? ? ?=1 Equation 3-45 To ensure the stationarity and invertibility of the above INARMA(1,1) process given by Equation 3-43, the two conditions of ?< 1 and ?< 1 must hold. The unconditional first and second moments of this process are: ?(??) =? 1 +? 1?? ??? Equation 3-46 var(??) = 1 1??2 ???+??+???2???+ (1 +? 2 + 2??)?? 2? Equation 3-47 The autocorrelation function (ACF) of this process is given by (see Appendix 3.A for the proof): M.Mohammadipour, 2009, Chapter 3 60 ?? INARMA ?1,1?= ? ??2 +?2?+?????2???+ (?+?? 2 +?2?+?)?? 2 ??+??+???2???+ (1 +?2 + 2??)?? 2 for ?= 1 ????1 for ?> 1 ? Equation 3-48 It can be seen that the ACF of an INARMA(1,1) dies exponentially, which is analogous to the ACF of the Gaussian ARMA(1,1) which is as follows (for ??= 0 and ?? 2 = 1): ?? ARMA (1,1) =? ?+?+?2?+??2 1 +?2 + 2?? for ?= 1 ????1 for ?> 1 ? Equation 3-49 3.3.8 INARMA(p,q) Model The INARMA(p,q) process is given by the following difference equation: ??=? ??????? ? ?=1 +??+? ??????? ? ?=1 Equation 3-50 where ??,???[0,1] and {??} is as before and thinning operations are performed independently of each other and also of the corresponding operations at previous times. The stationarity conditions of this process are the same as those of an INAR(p) process: to ensure that the above process is stationary, it is required that (?1,?2,?,??) are such that the roots of the p-order polynomial ? ???1? ??1??? ???1????= 0 lie inside the unit circle. Neal and Rao (2007) discuss the invertibility conditions for an INARMA(p,q) process for the moving average parameters (?1,?2,?,??). They assume that these conditions are the same as the those of an MA(q) process. However, they have not provided any proof in this regard and left it as an open question to investigate if this condition is sufficient for an INARMA(p,q) process to be invertible. M.Mohammadipour, 2009, Chapter 3 61 3.3.8.1 First and Second Unconditional Moments As mentioned before, the stochastic properties, including the autocorrelation function, of the general INARMA(p,q) process have not been found in the literature. Therefore, to answer the first research question ?How can the appropriate integer autoregressive moving average (INARMA) model be identified for a time series of counts??, here we investigate these properties. Obtaining the first unconditional moment of the INARMA process of Equation 3-50 is straightforward. It is given by: ?(??) =? 1 +? ?? ? ?=1 1?? ?? ? ?=1 ??? Equation 3-51 However, the derivation of the second unconditional moment is more challenging. We have found the unconditional variance of the INARMA process of Equation 3-50 as follows (see Appendix 3.B for the proof): var????= ?? 1?? ?? 2? ?=1 ? 1 +? ?? ? ?=1 1?? ?? ? ?=1 ? ??1??? ? ?=1 +? ??1??? ? ?=1 ? + ?? 2 1?? ?? 2? ?=1 ?1 +? ?? 2 ? ?=1 + 2 ? ??? min (?,?) ?=1 + 2? ? ???+??? ??? ?=1 ??1 ?=1 + 2? ? ??????? ??? ?=?+1 min (?,?) ?=1 1?? ?? 2? ?=1 Equation 3-52 where ?? ?? is the cross-covariance function derived in Appendix 3.C and ?? is the autocovariance at lag ? which can be expressed in terms of ?0 (or var????) from the equations obtained in the next section. It can be seen that if ?= 0, for an INAR(p) process, the unconditional variance would be: var????= 1 1?? ?? 2? ?=1 ?? ? ??1??? ? ?=1 1?? ?? ? ?=1 ???+?? 2 + 2? ? ???+??? ??? ?=1 ??1 ?=1 ? Equation 3-53 M.Mohammadipour, 2009, Chapter 3 62 In the above equation, ?? can be expressed in terms of ?0 based on the Yule-Walker equations of Equation 4-12. For example, for an INAR(2) process we have: ?1 =?1?0 +?2?1 and as a result: ?1 = ?1 1??2 ?0 The unconditional variance of an INAR(2) process can then be found from Equation 3-53 to be: var????= ??1??1 2 +?2?2?2 2??1?2 +?1 2?2 +?2 3???+ (1??1?2?2 +?1?2 +?2 2)?? 2 1??1??1 2 +?1 3?2?2 + 2?2 3??2 4 +?1?2 +?1?2 2??1?2 3 +?1 2?2 It can be seen from Equation 3-53 that the unconditional mean and variance of a PoINAR(p) process are not equal when the distribution of the innovations is Poisson. This has been also confirmed in the literature (Bu and McCabe, 2008). Also, when ?= 0, for an INMA(q) process, the variance would be: var????=?? ??1??? ? ?=1 ???+?1 +? ?? 2 ? ?=1 ??? 2 Equation 3-54 which agrees with the result found by Br?nn?s and Hall (2001) (Equation 3-38). Equation 3-52 is the first new result found in this PhD study. 3.3.8.2 Autocorrelation Function (ACF) The next step is finding the autocorrelation function of an INARMA(p,q) process. In order to do so, first, we need to find the covariance of this process. The covariance of INARMA(p,q) at lag ? is: ??= cov(??,????) According to the relation between ? and ? and ?, there are four cases: M.Mohammadipour, 2009, Chapter 3 63 ? ???, ??? ? ???, ?>? ? ?>?, ??? ? ?>?, ?>? Each of these cases will be considered in sequence. 1. If ???, ??? (? can be only equal to either ? or ? if ???) Figure ?3-1 The covariance at lag , , when qkpk ?? , ??= cov???,????? = cov???1????1 +?+???????+??+?1????1 +?+????????,????? = cov???1????1?,?????+?+ cov??????????,????? +cov??????????,?????+ cov????+1??????1?,?????+? + cov?????1?????+1?,?????+ cov??????????,????? ??=?1???1 +?+???1?1 +??var??????+??+1?1 +?+?????? +???0 ??+??+1?1 ??+?+???1????1??? ?? +?????? ?? where ?? ?? is the cross-covariance between ? and ? at lag ? (see Appendix 3.C). Then, the above equation can be written as: k k? ? ????? ??1 ?=1 ? ????? ? ?=?+1 ??var(????) ? ? 0 ? ? 0 ? ??????? 2? ?=?+1 ???? 2 ? ? M.Mohammadipour, 2009, Chapter 3 64 ??= ? ????? ??1 ?=1 +??var??????+ ? ???? ? ?=?+1 +???Z 2 + ? ?????? ?? ? ?=?+1 Note that if 2???, there will be a ?? in ? ???? ? ?=?+1 which has to be considered. This is because when ?= 2?, we have ?2??2???=?2???. 2. If ???, ?>? Figure ?3-2 The covariance at lag at lag , , when qk pk ?? , ??= cov???,????? = cov???1????1 +?+???????+??+?1????1 +?+????????,????? =?1???1 +?2???2 +?+???1?1 +??var??????+??+1?1 +??+2?2 +?+?????? Therefore, for the second case, the autocovariance at lag ? can be obtained from: ??= ? ????? ??1 ?=1 +??var??????+ ? ???? ? ?=?+1 Again, if 2???, there will be a ?? in ? ???? ? ?=?+1 which has to be considered. k k? ? ????? ??1 ?=1 ? ????? ? ?=?+1 ? ? 0 ??var(????) ? ? 0 ? ? M.Mohammadipour, 2009, Chapter 3 65 3. If ?>?, ??? Figure ?3-3 The covariance at lag , , when qk pk ?? , ??= cov???,????? = cov???1????1 +?+???????+??+?1????1 +?+????????,????? =?1???1 +?2???2 +?+??????+???0 ??+??+1?1 ??+?+???1????1??? ?? +?????? ?? Therefore, the autocovariance at lag ? for the third case can be obtained from: ??=? ????? ? ?=1 +???Z 2 + ? ?????? ?? ? ?=?+1 4. If ?>?, ?>? Figure ?3-4 The covariance at lag , , when qkpk ?? , k k? k k? ? ? 0 ? 0 ? ? ? ????? ? ?=1 ? 0 ? ? ? ????? ? ?=1 ? ? 0 ? ??????? 2? ?=?+1 ???? 2 ? ? M.Mohammadipour, 2009, Chapter 3 66 ??= cov???,????? = cov???1????1 +?+???????+??+?1????1 +?+????????,????? =?1???1 +?2???2 +?+??????=? ????? ? ?=1 Finally, the autocovariance at lag ? for the fourth case is given by: ??=? ????? ? ?=1 Therefore, we can write the autocorrelation function of an INARMA(p,q) process as follows: ?? = ? ? ? ? ? ? ? ? ? ? ? ? ? ????? ??1 ?=1 +?????+? ????? 2??1 ?=?+1 +? ????? ? ?=2?+1 +???? 2 +? ????? ??? ?=?+1 ?1??2????? for 2??? ? ????? ??1 ?=1 +?????+? ????? ? ?=?+1 +???? 2 +? ????? ??? ?=?+1 ??? for 2?>? ????,??? ? ? ? ? ????? ??1 ?=1 +?????+? ????? 2??1 ?=?+1 +? ????? ? ?=2?+1 ?1??2????? for 2??? ? ????? ??1 ?=1 +?????+? ????? ? ?=?+1 ??? for 2?>? ? ???,?>? ? Equation 3-55 where ??? is the second unconditional moment of the process given by the Equation 3-52 and ?? ?? is the cross-covariance given by the Equation 3.C-1 (see Appendix 3.C). The other two cases of ?>?,??? and ?>?,?>? are special cases of the above expressions. The above equation can be simply written as: ??=? ?1???1 +?2???2 +?+??????+???0 ??+??+1?1 ??+?+?????? ?? ?0 ??? ?1???1 +?2???2 +?+?????? ?>? ? Equation 3-56 Equation 3-56 is another new result of this research. Identification of the order of an INARMA(p,q) process requires both the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the process. The structure of the PACF of an INARMA(p,q) process will be discussed in chapter 4. M.Mohammadipour, 2009, Chapter 3 67 In order to test the Equation 3-56, we check if it results in the correct ACF for INAR(p) and INMA(q) processes. When ?= 0, i.e. for an INAR(p) process, it can be seen that the ACF based on the Equation 3-56 will be: ??=?1???1 +?2???2 +?+?????? which agrees with the result found by Du and Li (1991). When ?= 0, i.e. for an INMA(q) process, it can be seen that the ACF based on Equation 3-56 will be: ??=? ???? 2 +? ????? ??? ?=?+1 ?0 ??? 0 ?>? ? where ???=?? ??(1???) ? ?=1 ???+?1 +? ?? 2? ?=1 ??? 2. For an INMA(q) process with Poisson marginal distribution (??=?? 2 =?), the variance is given by: ???=?1 +? ?? ? ?=1 ?? It can be seen from Equation 3.C-1 that for an INMA(q) process ?? ??=??? for 0????, so we have: ??=? ??+? ????? ? ?=?+1 1 +? ?? ? ?=1 ??? 0 ?>? ? which agrees with the result in the literature (Br?nn?s and Hall, 2001). 3.3.9 Applications of INARMA Models Applications of INAR processes in the medical sciences can be found in, for example, Franke and Seligmann (1993) and Cardinal et al. (1999); and applications to economics in, for example, B?ckenholt (1999), Berglund and Br?nn?s (2001), Br?nn?s and Hellstr?m (2001), Rudholm (2001) and Freeland and McCabe (2004b). M.Mohammadipour, 2009, Chapter 3 68 Br?nn?s (1995) studies the consequences and required adaptations when explanatory variables are included in the INAR(1) model. He obtains new conditional least squares (CLS) and generalized method of moments estimators for the model containing explanatory variables and applies the INAR(1) model with Poisson marginal distribution to the number of Swedish mechanical paper and pulp mills during 1921-1981. INAR(1) models also have applications in inventory control. Aggoun et al. (1997) use an INAR(1) inventory model for perishable items, i.e. each item in the stock perishes in a given time period with an unknown probability. The sequence of these probabilities is assumed to be a homogeneous Markov chain and the paper finds the conditional probability distribution of this sequence and estimates the transition probabilities of the Markov chain. However, the model is not applied to real-world data. Cardinal et al. (1999) represent infectious disease incidence time series by INAR models. They state that real-valued time series models have been used in the analysis of infectious disease surveillance data, but argue that these models are not suitable in some cases such as the analysis of a rare disease. Meningococcal infection is considered as a rare disease in their study and the integer-valued INAR(5) model is fitted to the data set. B?ckenholt (1999) introduces the application of INAR models in investigating regularity and predictability of purchase behaviour over time. He uses a PoINAR(1) model for the analysis of longitudinal purchase data because there is a notion that purchase behaviour of nondurable goods is well-described by a Poisson process. The population of consumers is then divided to an unknown number of mutually exclusive and exhaustive segments, and within each a PoINAR(1) process is used to model the counts. The mixed PoINAR(1) model is finally applied to a powder detergent purchase data set of about 5000 households. Another straightforward application of INAR models can be found in a paper by Berglund and Br?nn?s (2001) in which they study the entry and exit of plants in Swedish municipalities as an INAR(1) process. In their model, they incorporate the variables affecting survival and entry and employ generalized method of moments M.Mohammadipour, 2009, Chapter 3 69 for estimation. Br?nn?s and Hellstr?m (2001) apply the INAR(1) model to the number of Swedish mechanical paper and pulp mills. They consider the number of firms in a region at a certain time to be equal to the number of firms surviving from the previous time period plus the number of new firms. They assume that the survival of a firm depends on the survival of other firms (dependent exits) and there is dependence between the survival of a firm and entry of a new firm (dependent entry and exit). The results for the set of data show that the correlation between exits is not significant, but that of entry and exit is significant. In another Swedish application, Rudholm (2001) analyses the factors affecting entry into the Swedish pharmaceuticals market using an INAR(1) model. Br?nn?s et al. (2002) find another application of INAR models in forecasting hotel guest nights which conventionally is based on economic demand models, pure time series analytical models, or on a mixture of them. They suggest that the daily number of guest nights for a specific hotel follows an INAR(1) process, and then proceed by cross-sectional and temporal aggregation of the model. The former means aggregation over more than two hotels which yields an INAR(1) model, and the latter means aggregation over time that results in an INARMA(1,1) model, later simplified as an INMA(1) model. Karlis (2002) introduces an INAR(1) model with a general mixed Poisson distribution for the innovation term which allows for overdispersion. The model is then applied to the number of forest fires in Greece in a two-month period (daily observations). Blundell et al. (2002) apply a Linear Feedback Model (LFM), which is derived from the INAR process, to the panel data of Hall et al. (1986). In their model, the technological output of a firm is a function of the corresponding R&D investment in current and previous periods, some unknown technology parameters, and the firm- specific propensity to patent. As another application, Gourieroux and Jasiak (2004) use an INAR(1) model to update premiums in car insurance and compare it to the standard negative binomial approach. Freeland and McCabe (2004b) use a PoINAR model for a monthly count data set of M.Mohammadipour, 2009, Chapter 3 70 claimants for wage loss benefit. They propose a method for producing coherent forecasts based on the conditional median rather than the conventional conditional mean. It is also argued that when the counts are low, the median should be accompanied by estimates of the probabilities associated with the point masses of the ?-step-ahead conditional distribution. A method for calculating confidence intervals for these probabilities is also presented. Quddus (2008) uses a PoINAR(1) model for analysis of traffic accidents in Great Britain. The results of his study show that an ARIMA model performs better than INAR(1) for geographically and temporally aggregated time series. However, as expected, the reverse is true for disaggregated low count time series (see section 3.3.10 for aggregation in INARMA models). It can be seen that INARMA models have been generally used for time series of counts (counts of objects, events, or individuals). Although this application area is based on the direct physical interpretation of the INARMA models, as suggested in the literature (McKenzie, 2003), this should not restrict these models to only such applications. This research is an attempt to use INARMA models beyond their direct interpretation to model intermittent demand. 3.3.10 Aggregation in INARMA Models Time series aggregation is a widely discussed subject for continuous-valued time series. It goes back over 50 years (Quenouille, 1958) and since then many papers have considered different aspects of aggregation in continuous-valued time series (see for example: Amemiya and Wu (1972), Brewer (1973), Harvey and Pierse (1984), Nijman and Palm (1990), Drost and Nijman (1993), Marcellino (1999), Teles and Wei (2002), and Man (2004)). Three types of aggregation have been identified in the literature which can be classified as: a. temporal aggregation b. cross-sectional aggregation c. over a forecast horizon aggregation M.Mohammadipour, 2009, Chapter 3 71 Temporal aggregation, also called flow scheme, refers to aggregation in which a low frequency time series (e.g. annual) is achieved from a high frequency time series (e.g. quarterly or monthly). It means that the low frequency variable is the sum of k consecutive periods of the high frequency variable. For example, the annual observations are the sum of the monthly observations every twelve periods. For Gaussian models, it has been proved that the aggregation of an AR(p) process produces an ARMA(p,q) process where ???. Jung and Tremayne (2003) provide a modified statistic based on the mean-variance equality property of the Poisson distribution. The modified statistic is given by: ??=?? ? ????1???(????) ? ?=2 ? (????)2 ? ?=1 Equation 4-3 which is asymptotically equivalent to the S-statistic under the null hypothesis. Here, again the null hypothesis is rejected if ??>??. 4.2.3 Portmanteau-type Tests Jung and Tremayne (2003) use two portmanteau-type tests originally designed by Venkataraman (1982) and Mills and Seneta (1989) to measure the goodness-of-fit in branching processes as tests for independence in a time series of counts. M.Mohammadipour, 2009, Chapter 4 89 Under the null hypothesis (?0) of no serial dependence or i.i.d. random variables, the modified version of the statistic presented by Venkataraman (1982) is: ???????= ? ??+1 2? ?=1 ?? (????) 2? ?=1 ? 2 ? ??????2(?????1??)2 ? ?=?+2 Equation 4-4 where ??=? ??????(??????) ? ?=?+1 /? (????) 2? ?=1 and ??1 is an arbitrary integer (in their Monte-Carlo study, Jung and Tremayne (2003) use ?= 1, 5, 10). Under the null hypothesis of i.i.d. Poisson variables {??}, ????(?) ? ??2(?) as ???. The second test is an adapted version of the test presented by Mills and Seneta (1989) with the statistic: ????????= ? ??+1 2? ?=1 ?? (????) 2? ?=1 ? 2 ? ??????2(?????1??)2 ? ?=?+2 Equation 4-5 It can be easily seen that this is exactly the same as ????(?) except that ??+1 is replaced by ??+1. Here, ?? is the kth order sample partial autocorrelation and ? is as defined before. Similarly, ?????(?) ? ??2(?) as ???. It should be noted that the first-order lag sample correlations are ignored in both statistics. These two tests can be used to distinguish between INAR(1) and INMA(1) structures. Two portmanteau tests used in the ARMA literature to find if the data has any serial dependence are the Box-Pierce and the Ljung-Box tests. The latter is an enhancement of the former to improve the performance of the test for small sample sizes (Ljung and Box, 1978). The Ljung-Box statistic is given by: ??=???+ 2?? ?? 2 ??? ? ?=1 Equation 4-6 where ? is the number of observations and ?? is the sample autocorrelation at lag ?. A large value of ?? indicates that the model is inadequate. M.Mohammadipour, 2009, Chapter 4 90 4.3 Identification based on ACF and PACF The sample autocorrelation function (SACF) and sample partial autocorrelation function (SPACF) have been widely used in the literature for identification of the autoregressive and moving average order of the INARMA models (Latour, 1998; Jung and Tremayne, 2006a; Zheng et al., 2006; Zhu and Joe, 2006; Bu and McCabe, 2008). In this section, we review these functions for different INARMA models. 4.3.1 Autocorrelation Function (ACF) The Autocorrelation Function (ACF) is defined as a plot of the autocorrelations at lag ? versus the lag ?. In this section, we investigate the autocorrelation function of an INARMA(p,q) process. First, we recall that the ACF of an INAR(p) process of: ??=?1????1 +?2????2 +?+???????+?? is determined by Du and Li (1991) as: ??=?1???1 +?2???2 +?+?????? Equation 4-7 It can be seen that the correlation structures of INAR(p) and AR(p) processes are the same. For an INMA(q) process of: ??=??+?1????1 +?2????2 +?+??????? it is shown by Br?nn?s and Hall (2001) that the ACF for an INMA(q) process with Poisson marginal distribution is given by: ??=? ? ????+? ??? ?=0 ? ?? ? ?=0 for ?= 1,?,? 0 for ?>? ? Equation 4-8 Again, it can be seen that this is analogous to the ACF of an MA(q) process. In chapter 3, we showed that the autocorrelation function of an INARMA(p,q) process is as follows: M.Mohammadipour, 2009, Chapter 4 91 ??=? ?1???1 +?2???2 +?+??????+???0 ??+??+1?1 ??+?+?????? ?? ?0 ??? ?1???1 +?2???2 +?+?????? ?>? ? Equation 4-9 When ???+ 1, the autocorrelation is: ??=?1???1 +?2???2 +?+?????? for ???+ 1 Equation 4-10 Therefore, the ACF of an INARMA(p,q) process is analogous to that of an ARMA(p,q) process and it can be used in identifying the integer-valued time series. 4.3.2 Partial Correlation Function (PACF) The correlation between two variables can be used as a measure of interdependence. However, when a variable is correlated with a second variable, this may be due to the fact that they both are correlated with another variable(s). Therefore, it may be of interest to examine the correlations between variables when other variables are held constant. These are called partial correlations (Hamilton, 1994). PACF is a device to identify the autoregressive order of a stationary time series. It has been shown that, although an AR(p) process has an infinite ACF, it can be described in terms of p non-zero functions of the autocorrelations (Box et al., 1994). In this section, we examine the partial autocorrelation function of INARMA processes. The section is organized as follows. First the PACF of an INAR(p) process is studied. The PACF of INMA(q) and INARMA(p,q) processes are then discussed. 4.3.2.1 PACF of an INAR(p) Model In this section, we examine the partial autocorrelation function of INAR processes. The INAR(p) process is defined by the recursion: M.Mohammadipour, 2009, Chapter 4 92 ??=?1????1 +?2????2 +?+???????+?? Here we assume that ?? has a Poisson distribution with parameter ?. Multiplying the above equation by ???? produces: ??????=????(?1????1) +????(?2????2) +?+????(???????) +?????? Equation 4-11 but we know that: ?(??) =?1?(???1) +?2?(???2) +?+???(????) +?(??) ?=???1 +?2 +?+???+? So, if we take the expected value of Equation 4-11 and subtract ?(??)?(????) from it, we have: ??=?1???1 +?2???2 +?+??????+ cov(????,??) ??0 Equation 4-12 considering the fact that ????????????????=???(????????) (Silva and Oliveira, 2004). The last term in the RHS is zero because ???? can only involve innovation terms up to time ??? and therefore is uncorrelated with ??. Dividing the Equation 4-12 by ?0 yields: ??=?1???1 +?2???2 +?+?????? ??0 Equation 4-13 which is analogous to the difference equation for Gaussian AR processes. The Yule- Walker equations are: ?1 = ?1 + ?2?1 + ? + ?????1 ?2 = ?1?1 + ?2 + ? + ?????2 ? ? ? ? ? ?? = ?1???1 + ?2???2 + ? + ?? Equation 4-14 The autocorrelation function of an INAR(p) process can be described in terms of ? M.Mohammadipour, 2009, Chapter 4 93 nonzero functions of the autocorrelations. In an integer autoregressive process of order ?: ??=??1???1 +??2???2 +?+??(??1)????+1 +??????? for ?= 1,2,?,? Equation 4-15 where ??? is the ?th coefficient in an integer autoregressive representation of order ?. Therefore ??? is the last coefficient and ???= 0 for ?>?. The Equation 4-15 leads to the Yule-Walker equations which may be written as: ? 1 ?1 ?2 ? ???1 ?1 1 ?1 ? ???2 ? ? ? ? ? ???1 ???2 ???3 ? 1 ?? ??1 ??2 ? ??? ?=? ?1 ?2 ? ?? ? Equation 4-16 As defined by Hamilton (1994), the ?th population partial autocorrelation (denoted by ???) is the last coefficient in a linear (for Gaussian ARMA processes) projection of ? on its ? most recent values. In the case of integer autoregressive models, it can be stated as: ??=??1????1 +??2????2 +?+???????? Equation 4-17 which results in the same sets of difference equations based on autocorrelations. The justification for using partial autocorrelations is that, if data really were generated by an INAR(p) process, only the ? most recent values of ? would be useful for forecasting and the coefficients on ?'s more than ? periods in the past are equal to zero, which means that: ???= 0 for ?=?+ 1,?+ 2,? We can express the INAR(p) process as follows: ??=????1????1??2????2??????????=???? ??????? ? ?=1 It can be seen that the series in the RHS of the above equation is finite. Therefore, the M.Mohammadipour, 2009, Chapter 4 94 PACF of an INAR (p) process is finite. Although an INAR(p) process differs from the AR(p) process due to the thinning operations, these two processes share some properties including the ACF and PACF structure. Therefore, using the sample partial autocorrelation function (SPACF) may help us in identifying the autoregressive order of an INAR time series. 4.3.2.2 PACF of an INMA(q) Model This section focuses on finding the partial autocorrelation function of an INMA process. The INMA(q) process has the following form: ??=??+?1????1 +?2????2 +?+??????? We again assume that ?? has a Poisson distribution with parameter ?. Using the same argument as for an INAR(p) process, it can be seen that the autocovariance function of an INMA(q) process is: ??=?[(??+?1????1 +?2????2 +?+???????). (????+?1??????1 +?2??????2 +?+?????????)] so ??=? (??+?1??+1 +?2??+2 +?+??????)?? 2 ?= 1,2,?,? 0 ?>? ? Equation 4-18 Thus, the autocorrelation function (ACF) is given by: ??=? ??+?1??+1 +?2??+2 +?+?????? 1 +?1 +?+?? ?= 1,2,?,? 0 ?>? ? Equation 4-19 It can be seen that the autocorrelation function of an INMA(q) process is zero, beyond the order ? of the process. As discussed earlier in section 4.3.2.1, if the data were generated by an INAR(p) M.Mohammadipour, 2009, Chapter 4 95 process, only the ? most recent values of ? would be useful for forecasting and therefore the PACF cuts off after lag ?. However, if the data were generated by an INMA(q) process with ??1, then the partial autocorrelation ??? asymptotically approaches zero instead of cutting off abruptly. The PACF of the process (???) can be obtained by solving the set of equations given by Equation 4-15 using the ACF of the process. For example, Al-Osh and Alzaid (1988) find the first and second partial autocorrelations of an INMA(1) process as: ?11 =?1 ?22 = ?2??1 2 1??1 2 = ??2 1 + 2? which can be seen is the same as that of a MA(1) process. 4.3.2.3 PACF of an INARMA(p,q) Model The partial autocorrelation function of an INARMA(p,q) process is examined in this section. This process satisfies the difference equation: ??=? ??????? ? ?=1 +??+? ??????? ? ?=1 An INARMA(p,q) process can be written in form of (see Appendix 4.A): ??=???? ? ??????? ?? ?=1 ? ?=1 Equation 4-20 where ??=? ?? ???? ? ?=1 + 1 0 ? ?. The values of ?? can be found by repeated multiplications of the parameters of INARMA(p,q) process. It can be seen that the series in the RHS of the Equation 4-20 is infinite. Therefore, the PACF of an INARMA(p,q) process, similar to an INMA(q) process, is infinite. M.Mohammadipour, 2009, Chapter 4 96 As shown in section 4.3.1, the structure of the autocorrelation function of the INARMA(p,q) process is analogous to that of an ARMA process. The structure of the partial autocorrelation function of this process is also similar to that of an ARMA process, i.e. the PACF of a mixed integer autoregressive moving average process is infinite and it has the same shape as the PACF of a pure integer moving average process. 4.4 Residual Analysis The residuals of the INARMA models can provide a check of model adequacy (Jung and Tremayne, 2006b). After identification of the appropriate model using ACF and PACF and estimation of the parameters of the identified model, the residuals should be examined to check for any serial dependence. Any dependence in the residuals would suggest that a different model should be used. The ACF and PACF of the residuals should be plotted for this reason. Freeland and McCabe (2004a) define two sets of residuals: one for the arrivals component and another for the continuation process of a PoINAR(p) process. The residuals for the continuation component are given by: ?1?=?????????????? for ?=?+ 1,?,? Equation 4-21 The residuals for the arrivals component are: ?2?=???? Equation 4-22 However, as they mention, these definitions are not practical because ??????? and ?? cannot be observed and should be replaced with their conditional expected values. Adding the new components then results in the usual definition of residuals for a PoINAR(p) process: ??=???? ????? ? ?=1 ?? Equation 4-23 M.Mohammadipour, 2009, Chapter 4 97 Bu and McCabe (2008) suggest that in order to check the adequacy of the selected model, checking the traditional residual of the Equation 4-23 is not enough and the components residuals (?1? and ?2?) should also be tested to examine the suitability of each component in the model. The expected value of ??????? and ?? are provided in terms of the conditional probabilities. This resolves the impracticality issue mentioned in Freeland and McCabe (2004a). The residuals of an INMA(q) process are given by (Br?nn?s and Hall, 2001): ??=?????? ????? ? ?=1 Equation 4-24 Based on Equation 4-23 and Equation 4-24, the residuals of an INARMA(p,q) model can be obtained from: ??=???? ????? ? ?=1 ???? ????? ? ?=1 Equation 4-25 It will be explained in section 4.5 that the identification methods used in this PhD thesis are not based on ACF and PACF. However, as an example, here we show what the ACF and PACF of one INAR(1) series look like. The series is selected from those of the 16,000 series data set of chapter 9. The time series plot for all 72 periods is provided in Figure 4-1. The sample ACF and sample PACF of the above series are presented in Figure 4-2. The sample PACFs suggest that an INAR(1) model is appropriate. The parameters of the identified INAR(1) model are then estimated using the Yule- Walker (YW) estimation method (see section 5.3.1) to be ?= 0.5581 and ?= 0.0552. The next step is to check the model?s adequacy using the residual analysis. The residuals of the INAR(1) model are defined as: ??=?????????1???? M.Mohammadipour, 2009, Chapter 4 98 Figure ?4-1 Time series plot of one demand series among 16,000 series Figure ?4-2 Correlograms of the selected series among 16,000 series where ??? and ??? are the Yule-Walker estimates of parameters in the INAR(1) model. If any dependence structure exists in the residuals, a different model specification example would be considered. In order to check if such dependence exists in our example, the SACFs and SPACFs of the residuals of the estimated INAR(1) model are depicted in Figure 4-3. The figure suggests that there is no obvious dependence structure left in the residuals. M.Mohammadipour, 2009, Chapter 4 99 Figure ?4-3 Correlograms of the residuals of the INAR(1) model 4.5 Identification based on Penalty Functions It has been argued that distinguishing between autoregressive moving average models based on the Box-Jenkins procedure is difficult (Chatfield and Prothero, 1973; Newbold and Granger, 1974). This is because the ACF and PACF plots cannot easily identify mixed ARMA models. Moreover, the identification of ARMA models usually involves subjective judgment. The same is true for identification of INARMA models using ACF and PACF. The Kullback-Leibler information (Kullback and Leibler, 1951) is used to measure the difference between two probability density functions ?(?) and ?(?): ???,??=?????log? ?(?) ?(?|?) ??? Equation 4-26 In the above equation, ???,?? denotes the information lost when ? is used to approximate ?. ? is considered to be fixed and ? varies over ?. Akaike (1973) introduces the Akaike information criterion (AIC) as an approximately unbiased estimate of Kullback-Leibler information. The AIC is given by: M.Mohammadipour, 2009, Chapter 4 100 AIC =??2?log?maximum likelihood?+ 2? Equation 4-27 where ? is the number of estimable parameters in the approximating model. Ozaki (1977) shows that the AIC of the ARMA(p,q) model is given by: AIC??log?? 2 + 2? Equation 4-28 where ?? 2 is the residual variance and ?=?+?+ 1. When the sample size is small (?/?< 40), the above expression is biased and the following bias correction is necessary (Hurvich and Tsai, 1989; Hurvich and Tsai, 1995): AICC ??log?? 2 + 2?+ 2?(?+ 1)/(????1) Equation 4-29 AIC has also been used in the INARMA literature (see for example: B?ckenholt, 1999; Brandt et al., 2000; Zhu and Joe, 2006). However, the complexity of the likelihood function of these models has led to some limitations in the use of AIC. The likelihood function of an INAR(p) process with Poisson innovations is given by (Bu, 2006): ???1,?,??,??= ? ?(??|???1,?,????) ? ?=?+1 Equation 4-30 where the conditional probability function ???????1,?,????? is: ???????1,?,?????= ? ? ???1 ?1 ??1 ?1?1??1? ???1??1 min????1 ,??? ?1=0 ? ? ???2 ?2 ??2 ?1?1??2? ???2??2 min????2 ,????1? ?2=0 ? ? ? ???? ?? ??? ??(1???) ??????? ?????????1+?+???1? ??????1 +?+???1??! min [????,?????1+?+???1?] ??=0 Equation 4-31 M.Mohammadipour, 2009, Chapter 4 101 It can be seen that the logarithm of Equation 4-31 cannot be simplified as in ARMA models. It should also be mentioned that the likelihood function of the INARMA(p,q) process is not established yet. 4.6 The Identification Procedure For the simulation experiment, we need an automated method for identification that can be used for thousands of replications. Because the likelihood function of an INARMA(p,q) process is not yet found and even the likelihood function of an INAR(p) process is very complicated, the penalty functions for these models are not easy to find. It has been shown in chapter 3 that an INAR process is analogous to an AR process in autocorrelation structure and also forecasting. Therefore, it has been suggested that the standard programmes for AR processes, which are mainly based on AIC or BIC, could also be used for INAR processes (Latour, 1998). The same is true for an INMA process regarding the ACF structure and the conditional expected value. As mentioned in the previous section, the use of AIC in the INARMA literature is limited to those processes for which likelihood functions have been derived. This excludes the INMA and mixed models. Based on the above argument by Latour (1998) we test the performance of AIC of Equation 4-28 (or where applicable, AICC of Equation 4-29) for INARMA models. As discussed in section 4.2, Jung and Tremayne (2003) suggest that in analysing the time series of counts, any serial dependence should first be detected. If no such dependence is found, the complicated INARMA methods can be replaced with easier methods for independent data. Based on the above argument, we use two identification procedures in this thesis: a two-stage and a one-stage method. In the two-stage method, the first stage distinguishes between the INARMA(0,0) and the other INARMA models. The Ljung-Box statistic of Equation 4-6 is used for this reason. This is because it is a standard test used for conventional ARMA models. Therefore, it is included in most software packages (including MATLAB which is used in this thesis) and, based on the argument by Latour (1998), it can be used for M.Mohammadipour, 2009, Chapter 4 102 INARMA models as well. It will be shown in chapter 8 that the rejection percentages under the null hypothesis of i.i.d. Poisson are comparable to the results of the tests suggested by Jung and Tremayne (2003). The AIC of Equation 4-28 is then used for identification among the other INARMA models. This is again based on the argument of Latour (1998) to use the standard programmes for ARMA models for INARMA models. It should also be mentioned that the AIC of ARMA models has been used in the INARMA literature (e.g. Br?nn?s and Quoreshi, 2004). The reliability of this identification procedure will be tested in the simulation chapter. To our knowledge, this has not been done in the literature before. The impact of misidentification on the forecast accuracy will also be checked. The one-stage method only uses the AIC of Equation 4-28 (or AICC of Equation 4-29) to identify the most appropriate model. The performance of these two methods will be compared. This will be done in terms of the percentage of time that the correct model is identified and also the accuracy of forecasts based on each method. 4.7 Conclusions In this chapter, the methods of identification of the autoregressive and moving average orders of an INARMA process have been reviewed. It has been shown that the autocorrelation and partial autocorrelation functions of an INAR(p) process have the same structure as those of an AR(p) process. The same is true for INMA(q) and INARMA(p,q) processes. Therefore, the estimates of the functions (SACF and SPACF) can identify the moving average and autoregressive orders, respectively. The residuals of the estimated INARMA process then need to be checked for any remaining correlations. Two identification procedures will be used in this thesis. A two-stage method is based on first using the Ljung-Box statistic to identify any correlation in the data series. The next step involves using the AIC of ARMA models to select from the other possible INARMA models. The performance of this procedure in terms of the percentage of time that the model is identified correctly and also the effect of M.Mohammadipour, 2009, Chapter 4 103 misidentification on forecast accuracy will be tested in chapter 8. To our knowledge, this has not been done in INARMA literature. The two stage method will then be compared to a one-stage identification method based on using the AIC to select among the INARMA models including INARMA(0,0). M.Mohammadipour, 2009, Chapter 5 104 Chapter 5 ESTIMATION IN INARMA MODELS 5.1 Introduction Having identified the appropriate INARMA model, we then need to estimate the parameters of the selected model. Different estimation methods have been used in the literature to estimate the parameters of INAR(p) and INMA(q) models, namely: ? Yule-Walker (YW) ? Conditional least squares (CLS) ? Maximum likelihood (ML) ? Generalized method of moments (GMM) Each of these methods is briefly reviewed. Table 5-1 lists the main studies on estimation of parameters of INARMA models. M.Mohammadipour, 2009, Chapter 5 105 Table ?5-1 Research papers on estimation of parameters of INARMA models Model YW CLS ML GMM INAR(1) Al-Osh and Alzaid, 1987 Al-Osh and Alzaid, 1987 Al-Osh and Alzaid, 1987 Br?nn?s, 1994 Br?nn?s and Hellstr?m, 2001 INAR(p) Du and Li, 1991 Jung and Tremayne, 2006b for INAR(2) Du and Li, 1991 Bu et al., 2008 INMA(1) Br?nn?s and Hall, 2001 Br?nn?s and Hall, 2001 Br?nn?s and Hall, 2001 INMA(q) Br?nn?s and Hall, 2001 for INMA(2) Br?nn?s and Hall, 2001 Br?nn?s and Hall, 2001 The YW method is based on using the Yule-Walker equations of: ?1 = ?1 + ?2?1 + ? + ?????1 ?2 = ?1?1 + ?2 + ? + ?????2 ? ? ? ? ? ?? = ?1???1 + ?2???2 + ? + ?? and replacing the theoretical autocorrelations ?? by the sample autocorrelations, ?? (Box et al., 1994): ??= ? ??????(??????) ? ?=?+1 ? ??????2 ? ?=1 Equation 5-1 Lawrence et al. (1978) develop the CLS estimation procedure for stochastic processes based on the minimization of a sum of squared deviations about conditional expectation. Maximum likelihood estimation method, as the name suggests, finds the parameters that maximize the likelihood of the sample data. The likelihood of the sample data is the probability of obtaining that particular set of data, given the specific probability distribution. In the generalized method of moments, a set of population moment conditions is first derived based on the assumptions of the model. Then the GMM estimates of the parameters are obtained such that these moment conditions are satisfied for the M.Mohammadipour, 2009, Chapter 5 106 sample data. This chapter is organized as follows. The estimate of the parameter of a Poisson INARMA(0,0) process is provided in section 5.2. The YW, CLS, CML, and GMM estimates for the parameters of INAR(1) and INAR(p) processes are reviewed in sections 5.3 and 5.4 (GMM only for INAR(1) process). The corresponding estimation methods for INMA(1) and INMA(q) processes are reviewed in sections 5.5 and 5.6. The YW and CLS estimates of the parameters of an INARMA(1,1) process are derived in section 5.7. Finding the ACF of an INARMA(p,q) model in chapter 3 enables us to find the YW estimates of these models. As an example, the YW estimates of an INARMA(2,2) model are derived in section 5.8. The conclusions are given in section 5.9. As will be discussed in chapter 7, four INARMA models are selected for simulation and empirical analysis to compete against the benchmark methods. These models are: INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1). The estimation methods used in this thesis for these models are CLS and YW for the last three and also CML for INAR(1). Therefore, these estimates are specifically given in this section. It will be discussed in section 5.2 that the CLS and ML estimation methods result in the same estimate for a Poisson INARMA(0,0) process. 5.2 Estimation in an INARMA(0,0) Model The INARMA(0,0) process with Poisson marginal distribution is simply an i.i.d. Poisson process of: ??=?? Equation 5-2 where ?? are i.i.d. Poisson random variables. The conditional expected value of ?? given ???1 is therefore given by: ???????1?=? Equation 5-3 where ? is the only parameter to be estimated. The conditional least squares estimate M.Mohammadipour, 2009, Chapter 5 107 of ? is obtained by minimizing the function: ?????=? [??????????1?] 2 ? ?=1 =? (????) 2 ? ?=1 Equation 5-4 with respect to ? for a sample of {?1,?2,?,??}. ?= ? ?? ? ?=1 ? Equation 5-5 The likelihood function of a sample of ? observations of an INARMA(0,0) process can be written as: ????=? ?????? ????! ? ?=1 Equation 5-6 The ML estimator of ? can be obtained by maximizing the log of the likelihood function in Equation 5-6. It can be seen that this results in the same estimator as that of CLS (Equation 5-5). 5.3 Estimation in an INAR(1) Model 5.3.1 YW for INAR(1) The Yule-Walker estimator for ? in an INAR(1) model was found by Al-Osh and Alzaid (1987) to be as follows: ?= ? ??????(??+1??) ??1 ?=0 ? ??????2 ? ?=0 Equation 5-7 where ? is the sample mean. Since ?? is assumed to have a Poisson distribution with parameter ?, the estimator for ? is: M.Mohammadipour, 2009, Chapter 5 108 ???= ? ?? ? ?=1 ? Equation 5-8 where ??=???????1. Jung and Tremayne (2006b) propose the same estimator for ?, but a slightly different estimator for ?, which is indicated by ???. They argue that, as the first order moment of the INAR(1) model is ?????= ? 1?? , ? can be estimated from: ???= (1??) ? ?? ? ?=1 ? Equation 5-9 In this thesis we use the Equation 5-7 and Equation 5-9 to obtain the YW estimates of an INAR(1) process. This is because the Equation 5-9 is based on observed data and not estimates of the innovations as in Equation 5-8. 5.3.2 CLS for INAR(1) It can be easily seen that in the INAR(1) model, ?? given ???1 is still a random variable due to the definition of the thinning operation. The conditional mean of ?? given ???1, which is the best one-step-ahead predictor (Br?nn?s and Hall, 2001), is: ???????1?=????1 +?=?(?,???1) Equation 5-10 where ?= (?,?)? is the vector of parameters to be estimated. Al-Osh and Alzaid (1987) employ a procedure developed by Klimko and Nelson (1978) and derive the estimators for ? and ? as follows: ?= ? ?????1 ? ?=1 ?(? ?? ? ?=1 ? ???1 ? ?=1 )/? ? ???1 2? ?=1 ?(? ???1 ? ?=1 ) 2/? Equation 5-11 and M.Mohammadipour, 2009, Chapter 5 109 ?=?? ?? ? ?=1 ??? ???1 ? ?=1 ?/? Equation 5-12 It can be easily verified that ?, ??/??, ??/??, ?2?/???? satisfy the regularity conditions proposed by Klimko and Nelson (1978). It follows that the CLS estimators are strongly consistent and asymptotically normally distributed as: ??????0?~?(?,??1??) where ?0 = (?0,?0)? denotes the true values of the parameters and: ??=?? ??(?0,???1) ??? . ??(?0,???1) ??? ? ?,?= 1,2 ??=???? 2(?0) ??(?0,???1) ??? . ??(?0,???1) ??? ? ?,?= 1,2 with ???? 0?= ????(? 0,???1). Freeland and McCabe (2005) show that the distributions of the CLS and YW estimators of a PoINAR(1) process are asymptotically equivalent. 5.3.3 CML for INAR(1) The Conditional Maximum Likelihood (CML) and Maximum Likelihood (ML) estimators for the PoINAR(1) process are provided by Al-Osh and Alzaid (1987). The likelihood function of a sample of ? observations from an INAR(1) process can be written as: ???,??=?(?1)? ?(??|???1) ? ?=2 Equation 5-13 where ?(??|???1) is given by: M.Mohammadipour, 2009, Chapter 5 110 ???????1?= ? ? ???1 ? ???(1??)???1?? ???????? ??????! min (???1 ,??) ?=0 Equation 5-14 Because the marginal distribution of the PoINAR(1) process is Poisson with mean ?/(1??), the unconditional likelihood function is: ???,??= ???/(1??)[ ? 1??] ?1 ??1?! ? ? ? ? ???1 ? ???(1??)???1?? ???????? ??????! min (???1 ,??) ?=0 ? ? ?=2 Equation 5-15 In order to find the conditional maximum likelihood estimation (CML), ?1 is assumed to be given and the conditional likelihood function is reduced to: ???,??=? ? ? ? ???1 ? ???(1??)???1?? ???????? ??????! min (???1 ,??) ?=0 ? ? ?=2 Equation 5-16 Then, the unconditional and conditional maximum likelihood estimators can be derived by maximizing the logarithm of the likelihood functions of Equation 5-15 and Equation 5-16, respectively. Al-Osh and Alzaid (1987) used the procedure of Sprott (1983) to eliminate one of the parameters in the derivatives of the log-likelihood function. ?log[???,??] ?? =? ?(?) ? ?=2 ????1?= 0 Equation 5-17 ?log[???,??] ?? =? ????????1????(?) ?(1??) ? ?=2 = 0 Equation 5-18 where ????=?????1?/?????. The Equation 5-18 results in: ?= ? ?? ? ?=2 ??? ???1 ? ?=2 ??1 Equation 5-19 M.Mohammadipour, 2009, Chapter 5 111 The Equation 5-19 can then be used in Equation 5-17 to find ?. The ML estimators of ? and ? have the following asymptotic distribution: ??? ??? ??? ?~?(?,??1) where the matrix ? is the Fisher information (Bu, 2006). Al-Osh and Alzaid (1987) compare the performance of YW, CLS and CML estimates in terms of bias and MSE in a simulation study. Their results suggest that, in general, CML is worth the extra effort because it has the least bias and MSE of all. However, for small sample sizes (??75) and small autoregressive parameter ??= 0.1?, because the sample contains many zero values, CML is not as good as YW in terms of bias and MSE. It is worth mentioning that their study only compares the accuracy of estimates and not their impact on forecast accuracy, which is done in this PhD thesis (see section 8.4). 5.3.4 Conditional GMM for INAR(1) Br?nn?s (1994) uses the conditional GMM estimation method of Hansen (1982) to estimate the parameters of a PoINAR(1) process. It is called a conditional GMM since the moment restrictions used are conditional. The GMM estimator is based on minimization of the function: ?=?(?)???1?(?) Equation 5-20 where ?= (?,?)? is the vector of the unknown parameters to be estimated and ?(?) is the vector of moment restrictions. When ? is the asymptotic covariance of ?(?), the GMM estimator is efficient. ? is first minimized based on using an identity matrix ? for ?. Then the estimates ? are used to from ?. The moment restrictions used are: 1 ? ? ?? ? ?=2 = 0 M.Mohammadipour, 2009, Chapter 5 112 1 ? ? ?(???1??) ? ?=2 = 0 where ?? is the one-step ahead prediction error, ??=????( ??|???1). Although the above moments are unconditional, they are equal to the conditional ones (Br?nn?s, 1994). It can be seen that when ? =?2, the GMM and the CLS are the same. Br?nn?s (1994) then compares the performance of CLS, ML, and GMM. The results show that for large values of ?, the GMM estimates have smaller MSE than CLS. In general, for large values of ?, ?GMM is close to ?ML in terms of MSE. But for small values of ?, ?ML outperforms ?GMM . Moreover, ?ML always has smaller MSE than ?GMM , although when ? increases this difference decreases. The results do not show any conclusive advantage by using GMM over ML in terms of bias. To conclude, we do not see any benefit in using GMM for an INAR(1) process, considering the fact that it does not outperform the maximum likelihood method, which is used in this research, in terms of MSE. 5.4 Estimation in an INAR(p) Model 5.4.1 YW for INAR(p) Du and Li (1991) generalize the Yule-Walker estimation method to estimate the parameters of an INAR(p) process. The Yule-Walker equations are: ??=? Equation 5-21 where ?= [?????]???, ?= (?1,?2 ,?,??)?, and ?= (?1,?2 ,?,??)?. Replacing the theoretical autocorrelations ?? by the sample autocorrelations ?? results in the YW estimate of ?. Then, ? can be estimated from the expected value of the INAR(p) process: M.Mohammadipour, 2009, Chapter 5 113 ?= (1??1?????)? Equation 5-22 These estimators are strongly consistent and asymptotically normally distributed (Silva and Silva, 2006). Also, for a large number of observations, YW estimators are very close to the CLS estimators (Du and Li, 1991). 5.4.2 CLS for INAR(p) The CLS estimators of an INAR(p) process are also derived by Du and Li (1991). The conditional expected value of the process is: ???????1,?,?2,?1?=?1???1 +?+??????+? Equation 5-23 The least squares criterion to be minimized is then: ?????= ? [??????????1,?,?2,?1?] 2 ? ?=?+1 Equation 5-24 where ?= (?1,?2,?,??,?) is the vector of parameters to be estimated. ? can be found by setting the partial derivatives to zero. ?????? ??? = 0 (?= 1,2,?,?) ?????? ?? = 0 Du and Li (1991) suggest that, for large samples, the CLS estimators for INAR(p) process are very close to Yule-Walker estimators. They also are strongly consistent and asymptotically normal. 5.4.3 CML for INAR(p) Bu et al. (2008) study the maximum likelihood estimators of a general INAR(p) M.Mohammadipour, 2009, Chapter 5 114 process. The conditional likelihood function of an INAR(p) process is: ???1,?2,?,??,??= ? ?(??|???1,?,????) ? ?=?+1 Equation 5-25 where the conditional probabilities are: ???????1,?,?????= ? ? ???1 ?1 ??1 ?1?1??1? ???1??1 min????1 ,??? ?1=0 ? ? ? ???2 ?2 ??2 ?1?1??2? ???2??2 min????2 ,????1? ?2=0 ??? ? ? ???? ?? ??? ??(1???) ??????? ?????????1+?+???1? ??????1 +?+???1??! min [????,?????1+?+???1?] ??=0 Equation 5-26 The asymptotic distribution of the maximum likelihood estimator is Normal: ???????~?(?,??1) where ?= (?1,?,??,?)? is the parameter vector and the matrix ? is the Fisher information. Bu et al. (2008) investigate the asymptotic benefit of ML over CLS for a PoINAR(2) process using the asymptotic relative efficiency (ARE) ratio between the two estimators, i.e. the ratio of their asymptotic variances. Their results show that for persistent processes (high values of ?1, ?2 or both), the ML estimates are asymptotically more efficient than the CLS estimates. For low values of ?1 and ?2 and also for high values of ?2 and low values of ?1, the benefit of ML over CLS is slight. They also compare the performance of ML and CLS in a simulation study for ?= 100 and ?= 500. The results suggest that there is a gain in terms of MSE in using ML for larger samples. For smaller samples, when ?1 and ?2 are small, CLS has lower MSE than ML. M.Mohammadipour, 2009, Chapter 5 115 5.5 Estimation in an INMA(1) Model 5.5.1 YW for INMA(1) The Yule-Walker estimator for ? in a PoINMA(1) process (??=??+?????1) is as follows: ?= ?1 1??1 Equation 5-27 where ?1 is the lag one sample autocorrelation given by the Equation 5-1. Then, ? can be estimated from the expected value of the process: ?= 1 1 +? ? ?? ? ?=1 ? Equation 5-28 Note that ? can also be estimated form the unconditional variance of the INMA(1) (Equation 3-28), but the estimator based on the unconditional mean has smaller variance (Br?nn?s and Hall, 2001). 5.5.2 CLS for INMA(1) The conditional expected value of ?? given ???1 for an INMA(1) process is given by: ???????1?=????1 +? Equation 5-29 The prediction error is: ??=???????1?? Equation 5-30 The CLS estimates of ? and ? can then obtained by minimizing the following function: ?????=? [???(????1 +?)] 2 ? ?=1 Equation 5-31 M.Mohammadipour, 2009, Chapter 5 116 with respect to ?, where ?= (?,?)? is the parameter vector to be estimated. The CLS estimates for ? and ? are: ?= ? ?????1 ? ?=1 ?(? ?? ? ?=1 ? ???1 ? ?=1 )/? ? ???1 2? ?=1 ?(? ???1 ? ?=1 ) 2/? Equation 5-32 ?=?? ?? ? ?=1 ??? ???1 ? ?=1 ?/? Equation 5-33 5.6 Estimation in an INMA(q) Model 5.6.1 YW for INMA(q) The autocorrelation function of an INMA(q) process of ??=??+?1????1 +?+ ??????q is given by the Equation 3-40. The ACF can be used to find the YW estimates of (?1,?,??). Once these parameters have been estimated, ? can be estimated from the expected value of the process: ?= 1 1 +? ?? ? ?=1 ? ?? ? ?=1 ? Equation 5-34 When the order of the INMA model increases, the equations to be solved become more complex. This is shown in Table 5-2. Table ?5-2 The relationship between the order of the model and the type of YW equation Model Equation INMA(1) Linear INMA(2) quadratic (2) INMA(3) quartic (4) INMA(4) sextic (6) M.Mohammadipour, 2009, Chapter 5 117 The results of Table 5-2 can be found by direct expansion of expressions in the Yule- Walker equations. As can be seen from the above table, for INMA processes with order higher than two, it becomes more complex to find the estimators and although such equations can be solved numerically, it is computationally expensive to find the Yule-Walker estimators for such processes. 5.6.2 CLS for INMA(q) The conditional first moment of the INMA(q) process is: ???????1?= ?+? ?????j ? ?=1 Equation 5-35 Hence, the forecast error is: ??=?????? ?????? ? ?=1 Equation 5-36 The least squares criterion is then: ?????= ? ??????? ?????? ? ?=1 ? 2 ? ?=?+1 Equation 5-37 The corresponding parameter vector ?= (?1,?,??,?)? for the minimum ????? can be obtained. 5.6.3 GMM based on Probability Generation Functions for INMA(q) In this estimation method, the probability generation functions (pgf) ?(?) and ??(?1,?2) are evaluated at any ?. Based on the law of large numbers: M.Mohammadipour, 2009, Chapter 5 118 ? (1???) ??? ?=1 ? ? ??(??) ? (1???) ??(1???) ????? ?=?+1 ??? ? ???(??,??) where ? ? denotes convergence in probability. Therefore, the moment conditions are formed as: ?1?= ? (1???) ??? ?=1 ? ??(??) ?2?,??= ? (1???) ??(1???) ????? ?=?+1 ??? ???(??,??) The GMM criterion to be minimized is then: ?=?? ?1? where ? is the vector of moment restrictions and ? is the covariance matrix of ?. Similar to section ?5.3.4, first it is assumed that ? is equal to the identity matrix. This results in a consistent and asymptotically normal estimator. However, if ? is known or a consistent estimator of ? is used, the GMM estimators would be more efficient than if ? is equal to ?. For a PoINMA(1) process, GMM estimators provided by Br?nn?s and Hall (2001) are as follows: ?= ?ln????ln?????1?ln????2 ?ln????ln?????1?ln????2(1??1) Equation 5-38 ?=? ln(?) ?1(1 +?) Equation 5-39 where ? and ? are the sample moments corresponding to ?(?1) and ?1(?1,?2), respectively. Br?nn?s and Hall (2001) compare the performance of YW, CLS and GMM for an M.Mohammadipour, 2009, Chapter 5 119 INMA(2) process. The results for the case that ?1 =?2 = 0.9 (note that the process is not invertible) suggest that GMM has the smallest bias and MSE among all estimators except for small sample sizes where CLS is the best. For all ?2 values except ?2 = 0.9, CLS is better than GMM in terms of MSE. Although they mention that for other values of ?1 and ?2 GMM has smaller bias and MSE in most cases, they did not present any results for further comparisons. Br?nn?s and Hall (2001) explain that the performance of the GMM estimator depends highly on the selection of ?-values. Finally, because of the better overall performance and its simplicity, they advocate the use of CLS instead of GMM. 5.7 Estimation in an INARMA(1,1) Model In this section, the YW and CLS estimators for the parameters of an INARMA(1,1) process, derived in this PhD research, are presented. 5.7.1 YW for INARMA(1,1) The ACF of an INARMA(1,1) process of ??=?????1 +??+?????1 is given by Equation 3-48. When the distribution of the innovations {??} is Poisson, the ACF is: ??=? ?+?+??+?2 + 2?2? 1 +?+?+ 3?? for ?= 1 ????1 for ?> 1 ? Equation 5-40 Hence, ? and ? can be estimated from: ?= ?2 ?1 = ? ??????(???2??) ? ?=3 ? ??????(???1??) ? ?=2 Equation 5-41 ?= ?1 +??(???1) ?1?1 + 3???1???2?2 Equation 5-42 M.Mohammadipour, 2009, Chapter 5 120 Then, ? can be estimated using the expected value of an INARMA(1,1) process: ?= 1?? 1 +? ? ?? ? ?=1 ? Equation 5-43 5.7.2 CLS for INARMA(1,1) The conditional expected value of an INARMA(1,1) process is: ?(??|???1) =????1 +?+????1 Equation 5-44 The conditional least squares criterion is therefore: ?????=? [????????1 +?+????1?] 2 ? ?=1 Equation 5-45 with ?= (?,?,?)? is the parameter vector to be estimated. The estimators for ?, ?, and ? can then be obtained by minimizing the above function with respect to ?. ?= ? ?2??????1????1 2 ?????????1????1 2 ????????1?????1? 2 +?????1????1??????1 ??2??????1????1???1 +????????1????1???1 ? ?2????1 2 ????1 2 ???????1?2????1 2 ??????1 2 ?????1?2 + 2?????1????1????1???1??2?????1???1?2 Equation 5-46 ?= ???????1???????1???1????????1 +?????1????1 ?????1 2 ??????1?2 Equation 5-47 ?= ?????????1??????1 ? Equation 5-48 where all the summations are from 1 to ? (see Appendix 5.A for the proof). M.Mohammadipour, 2009, Chapter 5 121 5.8 YW Estimators of an INARMA(2,2) Model The YW estimators of an INARMA(2,2) model can be obtained from the ACF of an INARMA(p,q) model derived in chapter 3. The INARMA(2,2) model has the form: ??=?1????1 +?2????2 +??+?1????1 +?2????2 Equation 5-49 From the Equation 3-52, the variance of the above process, when the innovations are Poisson distributed, the variance of an INARMA(2,2) process can be found as (see Appendix 5.B): var????= ? (1??1 2??2 2)(1??2)?2?1 2?2 ? ??1??2?? 1 +?1 +?2 1??1??2 ??1??1 2 +?2??2 2?+ 1 +?1 +?2 + 2?1?1 + 2?2?2?+ 2?1?2?1 + 2?1 2?2 + 2?1?1?2? Equation 5-50 The autocorrelation function of an INARMA(2,2) process can be found from the Equation 3-55 as: ??= ? ? ? ? ? ?1?0 +?2?1 +?1?+?2(?1 +?1)? ?0 ??? ?= 1 ?1?1 +?2?0 +?2? ?0 ??? ?= 2 ?1???1 +?2???2 ??? ?> 2 ? Equation 5-51 Based on the Equation 5-51, the autocorrelation of lags one to four can be used to estimate ?1, ?2, ?1, and ?2. ?1 =?1 +?2?1 + ?1?+?2(?1 +?1)? ?0 Equation 5-52 ?2 =?1?1 +?2 + ?2? ?0 Equation 5-53 ?3 =?1?2 +?2?1 Equation 5-54 M.Mohammadipour, 2009, Chapter 5 122 ?4 =?1?3 +?2?2 Equation 5-55 The last two equations can be used to find ?1 and ?2. ?1 = ?1?4??2?3 ?1?3??2 2 Equation 5-56 ?2 = ?3??1?2 ?1 Equation 5-57 where ?? is the sample autocorrelation at lag ? given by the Equation 5-1. Then, ?1 and ?2 can be found from Equation 5-52 and Equation 5-53. Finally, the expected value of the process can be used to estimate ?. ?=? 1??1??2 1 +?1 +?2 ?? Equation 5-58 where ?= ? ?? ? ?=1 ? . 5.9 Conclusions Different methods for estimating the parameters of INARMA models provided in the literature have been reviewed in this chapter. This includes YW, CLS, CML, and GMM. Not all these methods have been developed for all INARMA models. For example, the maximum likelihood function and therefore the CML estimators have been developed only for INAR(p) models. The performance of these estimators has been compared in some studies (Al-Osh and Alzaid, 1987; Br?nn?s, 1994; Br?nn?s, 1995; Br?nn?s and Hall, 2001; Bu et al., 2008). The results generally suggest that the CML is worth the extra effort especially for high values of ? (and ?1 for an INAR(2) process) and reasonably large samples. For lower values of ? and smaller samples, the CLS has lower MSE than ML. M.Mohammadipour, 2009, Chapter 5 123 The YW and CLS estimates for an INARMA(1,1) process are provided for the first time. The YW results are based on the ACF of an INARMA(p,q) model of Equation 3-55. These results along with YW, CLS, CML for INAR(1) process and YW and CLS for INMA(1) process will be used in simulation and empirical analyses of this PhD research (chapters 8 and 9). Finding the ACF of an INARMA(p,q) process in chapter 3 has enabled us to derive the YW estimators for these processes. As a further example, these estimators are obtained for an INARMA(2,2) process. We have decided not to follow the GMM method. This is because, for an INAR(1) process, it does not outperform the maximum likelihood estimator, which is covered by this study, in terms of MSE. For the INMA(1) process, no comparison has been done in the literature and the one that compared CLS and GMM for INMA(2) (Br?nn?s and Hall, 2001) does not provide results for all parameter sets. It is worth mentioning that all of the above-mentioned studies only compared the bias and MSE of the estimates and not their impact on forecast accuracy. This will be done in our simulation experiment (see section 8.4). M.Mohammadipour, 2009, Chapter 6 124 Chapter 6 FORECASTING IN INARMA MODELS 6.1 Introduction Having discussed some stochastic properties of INARMA models, and identification and estimation of the parameters of these models, we now investigate how these models can be used in forecasting future values of an observed time series. This section is organized as follows. The minimum mean square error (MMSE) forecasts for INAR(p), INMA(q) and INARMA(p,q) processes are reviewed in section 6.2. The lead time aggregation and forecasting of INARMA processes is then discussed in section 6.3. Results on lead time aggregation and forecasting for INAR(1), INMA(1), and INARMA(1,1) processes are presented. The conclusions are given in section 6.4. M.Mohammadipour, 2009, Chapter 6 125 6.2 MMSE Forecasts The most common forecasting procedure discussed in the time series literature is using the conditional expectation (Freeland and McCabe, 2004b). The main advantage of this method, apart from being simple, is that it produces forecasts with minimum mean square error (MMSE). Freeland and McCabe (2004b) argue that this method does not produce coherent forecasts for INARMA models. Coherency means that forecasts should comply with time series restrictions, in this case being integers. Freeland and McCabe (2004b) suggest the median of the distribution and use the ?-step-ahead conditional distribution to produce coherent forecasts for the PoINAR(1) model. Bu and McCabe (2008) present a procedure to produce h-step-ahead distribution forecasts for the PoINAR(p) process using the transition probability function of the process. Jung and Tremayne (2006b) introduce a Monte Carlo procedure to estimate the ?-step-ahead forecast distribution for INAR(1) and INAR(2) processes. This PhD research tries to apply INARMA models for intermittent demand forecasting. We are especially interested in comparing the accuracy of forecasts produced by INARMA methods to non-optimal smoothing-based methods (the last research question, p.7). For this reason, we compare the point forecasts of all methods using the accuracy measures suggested in section 2.4.3 (including MSE). We focus on the conditional expectation since it provides the MMSE forecasts for INARMA methods. It will be discussed in chapter 10 that forecasting the whole distribution can be considered as a further research avenue. 6.2.1 MMSE Forecasts for an INAR(p) Model Minimum mean square error (MMSE) forecasts are used to find ??+?, ?= 1,2,?,? of the process ?? based on the observed series of {?1,?,??}. The MMSE forecast of the process is given by: ??+?=????+???,?,?1? Equation ?6-1 M.Mohammadipour, 2009, Chapter 6 126 As indicated by its title, this method yields forecasts with minimum MSE. For an INAR(p) model of Equation 3-19, we have: ??+?=?1??+??1 +?2??+??2 +?+????+???+? Equation 6-2 where the ? values on the RHS of Equation 6-2 may be either actual or forecast values (Du and Li, 1991; Jung and Tremayne, 2006b). This is shown in Figure 6-1 for the case where ???. Figure ?6-1 h-step-ahead forecast for an INAR(p) model when ph ? This is called using a single model for all horizons. For example, for an INAR(2) process, the h-step ahead forecast is given by: ??+?=?1??+??1 +?2??+??2 +? Equation 6-3 This implies that for large ?, the forecasts converge to the unconditional mean of the INAR(2) process that is: ??+?? ? 1??1??2 Some authors suggested that using different models for different horizons can improve forecast accuracy (Cox, 1961; Tiao and Xu, 1993; Kang, 2003). For an AR(p) model, this is: ??+?=?1,???+?2,????1 +?+??(?),????????+1 +??,? Equation 6-4 ??+? ??+??1 ??+1 ? ?? ???1 ??+??? ? ??1 non-observed terms ???+ 1 observed terms M.Mohammadipour, 2009, Chapter 6 127 It can be seen from Equation 6-4 that even the order of the AR model depends on the forecast horizon ?. 6.2.2 MMSE Forecasts for an INMA(q) Model As studied by Br?nn?s and Hall (2001), for an INMA(q) process of Equation 3-36, the MMSE one-step ahead forecast can be obtained from: ??+1 =?1??+?2???1 +?+??????+1 +? Equation 6-5 The forecast error variance is: var(??) =??1 +? ??(1???) ? ?=1 ? Note that when ? is a random variable, var?????=?2var???+??1????(?), but when ? is given as in the above case, var?????=??1????(?). The h-step ahead forecast when ??? is given by: ??+?=????+?+????+???+?(1 +?1 +?+???1) Equation 6-6 This is shown in Figure 6-2. In the above equation, the ? values on the RHS can be estimated from the previous estimated ?s and observed ?s based on Equation 3-36. Figure ?6-2 h-step-ahead forecast for an INMA(q) model when qh ? ??+? ?(??+??1) =? ?(??+1) =? ? ?? ???1 ??+??? ? ??1 non-observed terms ???+ 1 observed terms M.Mohammadipour, 2009, Chapter 6 128 The forecast error variance for ??? is: var(??) =??1 +? ?? ??1 ?=1 +? ??(1???) ? ?=? ? When ?>?, the ?-step ahead forecast becomes: ??+?=??1 +? ?? ? ?=1 ? Equation 6-7 with the forecast error variance of var(??) =??1 +? ?? ? ?=1 ?. 6.2.3 MMSE Forecasts for an INARMA(p,q) Model The above results can be generalized for an INARMA(p,q) process. The MMSE one- step-ahead forecast is then: ??+1 =?1??+?+??????+1 +?+?1??+?+??????+1 Equation 6-8 The h-step ahead forecast when ??? will be: ??+?=?1??+??1 +?+????+???+?+????+?+????+???+?(?1 +?+???1) Equation 6-9 where ? values on the RHS of Equation 6-9 may be either actual or forecast values. When ?>?, the h-step ahead forecast becomes: ??+?=?1??+??1 +?+????+???+?? ?? ? ?=0 Equation 6-10 where again ? values on the RHS of the above equation may be either actual or forecast values and ?0 = 1. M.Mohammadipour, 2009, Chapter 6 129 6.3 Forecasting over Lead Time In this section, forecasting over a lead time is discussed. Lead time forecasting has applications in many areas, particularly in an inventory management context, where forecasts are needed over the period that it takes from placing an order to receiving it from the supplier. It can be easily seen that for an INARMA(0,0) process, the lead time aggregated process is: ? ??+? ?+1 ?=1 =??+1 +??+2 +?+??+?+1 =? ??+? ?+1 ?=1 Therefore, the conditional expected value and variance of the above equation are: ??? ??+? ?+1 ?=1 |???= var?? ??+? ?+1 ?=1 |???= (?+ 1)? which is expected as the aggregated process is the sum of (?+ 1) independent Poisson random variables which is in fact a Poisson variable with parameter (?+ 1)?. This section is organized as follows. First, the results of over-lead time aggregation and forecasting of INAR(1) and INMA(1) processes are presented. These results, along with similar results for INAR(2) and INARMA(1,2) processes in Appendices 6.A and 6.B, will then help us to find the over-lead time aggregation of the INARMA(p,q) process. The corresponding results for an INARMA(1,1) process are also provided, which will be used in chapters 8 and 9. 6.3.1 Lead Time Forecasting for an INAR(1) Model For the INAR(1) process of ??=?????1 +??, the cumulative ? over lead time ? is given by: ? ??+? ?+1 ?=1 =??+1 +??+2 +?+??+?+1 M.Mohammadipour, 2009, Chapter 6 130 =?????+??+1?+?? 2???+????+1 +??+2? +?+???+1???+? ????+1 +? ??1???+2 +?+??+?+1? Equation 6-11 Because ???+????(?+?)??, the above equation can be written as: ? ??+? ?+1 ?=1 =? ? ?? 1 ??? ?? 1 ?=1 ?+1 ?=1 +? ? ?? 2 ???+?? ?? 2 ?=1 ?+1 ?=1 Equation 6-12 where ?? 1 is the number of ?? terms in each of {??+?}?=1 ?+1 in Equation 6-11, ?? 1 is the corresponding coefficient for each ??, ?? 2 is the number of ??+?? terms in each of {??+?}?=1 ?+1 in Equation 6-11, and ?? 2 is the corresponding coefficient for each ??+??. All of these terms are explained below. It can be seen that because the process is an integer autoregressive of order one, each of {??+?}?=1 ?+1 yields only one ?? in Equation 6-11; therefore, ?? 1 = 1. The corresponding coefficient for ?? in each of {??+?}?=1 ?+1 (say ??+2) is obtained from ? thinned the coefficient of ?? in the previous term (in this case ??+1). As a result, ?? 1 =??. These coefficients are shown in Table 6-1. Table ?6-1 Coefficients of in each of for an INAR(1) model ?= 1,?= 1 ?11 1 =? ?= 2,?= 1 ?12 1 =?2 ? ? ?=?+ 1,?= 1 ?1(?+1) 1 =??+1 It can be seen from Equation 6-11 that due to the repeated substitution of ??+?, the number of ??+?? increases in each of {??+?}?=1 ?+1 . This number, shown by ?? 2, can be obtained from ???1 2 + 1. This means that each of {??+?}?=1 ?+1 (say ??+2) has one more ? compared to the previous one (which is ??+1 in this case). The corresponding coefficient for each ??+??, shown by ?? 2 , is ? thinned the corresponding coefficient in the previous term (???????1? 2 ). ?+?? is the subscript of innovation terms in tY 11 ? ?? l jjtY }{ M.Mohammadipour, 2009, Chapter 6 131 each of {??+?}?=1 ?+1 and from Equation 6-11 it can be easily seen that ?? is given by Equation 6-13. All of these terms are shown in Table 6-2. ??=? ??(??1) for 1??????1 2 ? for ???1 2 ??(??1) ? ?? ?= ? ? ? ? ? ? ? ? ? ? ? ? ????(???) ? ?= 1,?,???? ? ? ? ?1??(??1) ? ?=???2 ? + 1,?,???2 ? +???1 ? ??+(??1) ?=???1 ? + 1 ?????(??1) ? ????(???) ? ?= 1,?,???? ? ? ? ?1??(??1) ? ?=???2 ? + 1,?,???2 ? +???1 ? ??>??(??1) ? ?? ?+1 =?? ???? ?+1? ?=1 ?+ (?+ 1) ?? ?+1 = ? ? ? ? ? ????(???) ?+1 ?= 1,?,???? ?+1 ? ? ?1??(??1) ?+1 ?=???2 ?+1 + 1,?,???2 ?+1 +???1 ?+1 ??,?,?1 , 1 ?=???1 ?+1 + 1,?,???1 ?+1 +?? ?+1 ? ??= ? ? ? ? ? {??(???)} ?= 1,?,???? ?+1 ? ? {??(??1)} ?=? ???? ?+1? ?=2 + 1,?, (? ???? ?+1? ?=2 ) +???1 ?+1 ???,?,??1,? ?=? ???? ?+1? ?=1 + 1,?,?? ?+1 ? Now, in order to find the forecast over lead time, we need to calculate the expected value of the aggregated process given the p-previous observations. ??? ??+? ?+1 ?=1 |????+1,?,???1 ,???=?? ? ?? 1 ?? 1 ?=1 ?+1 ?=1 ???+?? ? ?? 2 ?? 2 ?=1 ?+1 ?=1 ????1 +? +?? ? ?? ? ?? ? ?=1 ?+1 ?=1 ?????+1 +?? ? ?? ?+1 ?? ?+1 ?=1 ?+1 ?=1 ?? Equation 6-27 M.Mohammadipour, 2009, Chapter 6 136 6.4 Conclusions Forecasting with an INARMA process is discussed in this chapter. The minimum mean square error (MMSE) forecasts for INAR(p), INMA(q) and INARMA(p,q) processes are reviewed. This includes both one-step and h-step ahead forecasts. These forecasts are based on the conditional expected value of the process and, as argued by McCabe and Martin (2005), these are not coherent forecasts. This means that the results are not necessarily integers. However, using the conventional forecasting method of conditional expectations is the most widely used approach in the literature even for count series. It is shown in this chapter that the aggregation of an INARMA(p,q) process with Poisson innovations (with mean ?) over a lead time ? results in an INARMA(p,q) process with the same autoregressive and moving average parameters and the innovation parameter of (?+ 1)?. The lead time aggregation and forecasting for the INARMA(p,q) process is obtained. In order to understand the implications of the results, some examples including a range of autoregressive and moving average processes are provided. It will be discussed in chapter 7 that four INARMA models will be used in simulation and empirical analysis of this thesis. These models are INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1). Therefore, the lead time forecast for the last three processes are presented in this chapter. The lead time forecast of an INARMA(0,0) can simply be obtained from ??+ 1??. M.Mohammadipour, 2009, Chapter 7 137 Chapter 7 SIMULATION DESIGN 7.1 Introduction This chapter addresses a simulation experiment based on theoretically generated data. A model-based simulation shows the evolution through time of a stochastic process, represented by a mathematical model through multiple realizations of the process. In this research, simulation is used for various reasons including: ? to assess the effects of the approximations made for the mathematical model ? to test the performance of identification methods ? to measure the accuracy of estimates of the model?s parameters ? to assess the sensitivity of forecast accuracy to control parameters such as the number of observations and the sparsity of data (based on the INARMA parameters) M.Mohammadipour, 2009, Chapter 7 138 ? to compare the forecasts of the mathematical model with other benchmark methods. The chapter is organized as follows. The reasons for conducting simulation are discussed in section 7.2. The simulation design is defined in section 7.3, including the range of INARMA models to be used in the simulation, the control parameters and the performance metrics. Verification of the simulation is discussed in section 7.4 and, finally, section 7.5 provides the conclusions. 7.2 Rationale for Simulation In chapter 4, two identification methods were discussed, namely the two-stage and the one-stage methods. Simulation enables us to find the percentage of theoretically generated INARMA time series that can be identified correctly by each of these methods. A further application of the simulation model is to investigate the effect of identifying an incorrect model for a specific series, or misidentification, on the accuracy of forecasts. The next step in the INARMA methodology is estimating the parameters of the identified model. As explained in detail in chapter 5, Conditional Least Squares (CLS) and Yule-Walker (YW) are the two estimation methods used (CML will also be used for the INAR(1) process). The role of simulation is to compare the results of these methods in terms of: (i) how close are the estimates to the real parameters, which are known when theoretically generated data are being used, and (ii) which estimation method results in better forecasts. The simulation model will be based on the assumption that the distribution of the innovations is Poisson. Although other distributions have been proposed in the literature including compound Poisson (McKenzie, 2003), negative Binomial (McKenzie, 1985; Al-Osh and Alzaid, 1987; Br?nn?s and Hall, 2001) and the Geometric (McKenzie, 1986; Alzaid and Al-Osh, 1988), this research only focuses on the Poisson. The sensitivity of the results to the distributional assumption can be analyzed but this will not be covered in this thesis. Other marginal distributions are beyond this research?s scope. The Poisson distribution is probably the most commonly used distribution in modelling counting processes (Alzaid and Al-Osh, M.Mohammadipour, 2009, Chapter 7 139 1990). It is the only distribution among the class of discrete self-decomposable1 distributions which has a finite mean (Silva and Oliveira, 2004). Another property of interest is that in the INAR(1) and INMA(q) processes, the Poisson distribution plays a role similar to that of the Gaussian distribution in the AR(1) process. However, Jung and Tremayne (2006b) argued that only an INAR(2)-AA process with Poisson innovations results in a process with Poisson marginal distribution and the same is not true for an INAR(2)-DL process. Another advantage of the Poisson over other distributions is that it has only one parameter to estimate. One of the main concerns in forecasting intermittent series is the length of available data history. This is because in practice we may be limited by short length of history. For example, the 3,000 series that we use in empirical analysis (see chapter 9) only has 24 periods of monthly data. Simulation enables us to check the sensitivity of the identification, estimation, and forecasting results to the length of the series. Once the forecasting results have been established, simulation can be used to compare these results with benchmark methods, Croston, SBA, and SBJ methods (see chapter 2 for detailed discussion on benchmark methods). This includes one-step ahead, ?-step ahead, and lead time forecasts. In a nutshell, simulation is conducted to analyze the sensitivity of results to: the sparsity of data, the length of history, the parameters? ranges, the estimation methods, and the effect of misidentification. It also enables us to compare the INARMA forecasts with those of benchmark methods using different accuracy measures. 7.3 Simulation Design 7.3.1 The Range of Series Different integer autoregressive moving average processes will be used to test the 1 A distribution with probability generating function (p.g.f) ? is called discrete self-decomposable if: ????=??1??+?????(?) |?|?1 ??(0,1) where ?? is a p.g.f. The above equation can also be written in the form of: ?=????+?? where ???? and ?? are independent and ? ? is distributed as ? (Sueutel, F. W. and K. van Harn (1979). Discrete analogues of self-decomposability and stability. Annals of Probability 7(5): 893- 899.). M.Mohammadipour, 2009, Chapter 7 140 mathematical findings. We consider an INAR process, an INMA process and a mixed INARMA process. In order to test the performance of the benchmark methods, the special case of INARMA(0,0) (or simply an i.i.d. Poisson process) is also used. Therefore, the following four processes are assumed for this study: INARMA(0,0), INARMA(1,0), INARMA(0,1), INARMA(1,1). An extension to this study would be to examine higher order INARMA processes. However, as shown in a later chapter, the simpler models ((0,0) and (1,0)) perform very well on empirical data. Using the above models also has the benefit of having few parameters to be estimated. 7.3.2 Producing INARMA(p,q) Series Since it has been assumed that the innovations are Poisson distributed (??~???(?)), we first need to generate i.i.d. Poisson random numbers. The simulation code is written in MATLAB 6.1. Hence, we use the poissrnd function from MATLAB?s statistics toolbox. The performance of this function is tested by the Poisson dispersion and the score tests (see section 4.2.2) and the results confirm the accuracy of the function. Next, by assuming the values of autoregressive parameters {??}?=1 ? and moving average parameters {??}?=1 ? , the autoregressive and moving average components are generated using a Binomial random number generator. This is because, based on the properties of binomial thinning discussed in chapter 3, ??? given ? has a binomial distribution with parameters (?,?). Therefore, for example in an INAR(1) model (??=?????1 +??), ?????1 is obtained from generating a random Binomial number with parameters (???1,?). The Binomial numbers are generated using the binornd function from the MATLAB?s statistics toolbox as a sum of Bernoulli random variables. The performance of this function is also tested by the score test and a built-in goodness-of-fit test (based on chi-square). The results, again, support the use of this function. Then, the INARMA series is generated from the model: M.Mohammadipour, 2009, Chapter 7 141 ??=? ??????? ? ?=1 +??+? ??????? ? ?=1 Equation 7-1 In order to obtain a stationary series, the series is initialized with the expected value of each process (Al-Osh and Alzaid, 1987; Br?nn?s, 1994). The expected value of the above process is given by: ?????= ??1 +? ?? ? ?=1 ? 1?? ?? ? ?=1 Equation 7-2 7.3.3 Control Parameters The control parameters of the simulation are: the mean of the Poisson innovations (?), autoregressive and moving average parameters ({??}?=1 ? , {??}?=1 ? ), the length of the series (?), the forecast horizon (?), the length of the lead time (?), and the benchmark methods? parameters. In this section, the ranges of these control parameters are reviewed. 7.3.3.1 INARMA Parameters From the definition of the thinning operation it is obvious that the autoregressive and moving average parameters represent the chance of surviving for elements of the process at time ??1 (????s and ????s, respectively). Therefore, these parameters are probabilities and can only take values in the range [0,1]. Other restrictions have to be applied on the autoregressive and moving average parameters in order to assure the stationarity and invertibility of the process. Table 7-1 reviews the range of values that these parameters can take for the INARMA processes selected in section 7.3.1 (see section 3.3.8 for stationarity and invertibility conditions of an INARMA(p,q) process). M.Mohammadipour, 2009, Chapter 7 142 Table ?7-1 Range of autoregressive and moving average parameters INARMA(p,q) models Range of autoregressive parameters Range of moving average parameters INARMA(0,0) - - INARMA(1,0) 0??< 1 stationarity condition: ??1 - INARMA(0,1) - 0??< 1 invertibility condition: ??1 INARMA(1,1) 0??< 1 stationarity condition: ??1 0??< 1 invertibility condition: ??1 The range of parameters for some simulation studies reported in the literature are reviewed in Table 7-2. Table ?7-2 Range of INARMA parameters studied in the literature Study Number of observations ? AR or MA parameter ? or ? Innovation parameter ? Number of replications Al-Osh and Alzaid (1987) ?= 50, 75, 100, 200 ?= 0.1, 0.2, 0.9 ?= 1 ?0.5?, 3 200 Br?nn?s and Hall (2001) ?= 10 ?10?, 100 ?100?, 500 ??= 0.1, 0.5, 0.9 ?= 5 1000 Br?nn?s and Hellstr?m (2001) ?= 50 , 100 , 200 ?= 0.5, 0.7, 0.9 ?= 5, 10 1000 Silva and Oliveira (2004) ?= 64, 128, 512, 1024 ?= 0.1, 0.5, 0.9 ?= 1, 3 200 Silva et al. (2005) ?= 25, 50, 100 ?= 0.1, 0.3, 0.7, 0.9 ?= 1, 3 Bu et al. (2008) ?= 100, 500 ??= 0.1, 0.3, 0.5, 0.7 ?= 1 1000 Based on the constraints of Table 7-1, and taking into account previous experiments (Table 7-2), the parameter space for the four selected INARMA models used in this thesis is shown in Table 7-3. If the discrete variates are large numbers, they can be approximated by continuous variates. It is when they are relatively small integers that using integer autoregressive moving average models becomes justifiable (McKenzie, 2003). Therefore, the innovation term (?) has to be defined to assure the observations are small integers. As can be seen from Table 7-3, we assume a range of ?= [0.5,5] for most models. For INARMA(0,0) we consider two other values of ?= 0.3 and ?= 20. This is to test the Croston-SBA categorization for highly intermittent and barely intermittent series. M.Mohammadipour, 2009, Chapter 7 143 Table ?7-3 Parameter space for the selected INARMA models INARMA(p,q) models Parameters INARMA(0,0) ?= 0.3, 0.5, 0.7, 1, 3, 5, 20 INARMA(1,0) ?= 0.1,?= 0.5 ?= 0.5,?= 0.5 ?= 0.9,?= 0.5 ?= 0.1,?= 1 ?= 0.5,?= 1 ?= 0.9,?= 1 ?= 0.1,?= 3 ?= 0.5,?= 3 ?= 0.9,?= 3 ?= 0.1,?= 5 ?= 0.5,?= 5 ?= 0.9,?= 5 INARMA(0,1) ?= 0.1,?= 0.5 ?= 0.5,?= 0.5 ?= 0.9,?= 0.5 ?= 0.1,?= 1 ?= 0.5,?= 1 ?= 0.9,?= 1 ?= 0.1,?= 3 ?= 0.5,?= 3 ?= 0.9,?= 3 ?= 0.1,?= 5 ?= 0.5,?= 5 ?= 0.9,?= 5 INARMA(1,1) ?= 0.1,?= 0.1,?= 0.5 ?= 0.1,?= 0.9,?= 0.5 ?= 0.5,?= 0.5,?= 0.5 ?= 0.9,?= 0.1,?= 0.5 ?= 0.1,?= 0.1,?= 1 ?= 0.1,?= 0.9,?= 1 ?= 0.5,?= 0.5,?= 1 ?= 0.9,?= 0.1,?= 1 ?= 0.1,?= 0.1,?= 5 ?= 0.1,?= 0.9,?= 5 ?= 0.5,?= 0.5,?= 5 ?= 0.9,?= 0.1,?= 5 7.3.3.2 Length of Series Different lengths of series are considered in order to test the sensitivity of results (identification, estimation, and forecasts accuracy) to the length of history. Because in real cases, we are often restricted by the short lengths of history (as will be seen in empirical analysis of this thesis) we use ?= 24, 36, 48, 96. Only for investigating the accuracy of estimates in terms of bias and MSE (section 8.3), ?= 500 is also added to the above cases. The first half of the observations is assigned for identification and estimation, and is referred to as the estimation period. This also includes the benchmark methods of Croston, SBA and SBJ. The second half is left for forecasting and is called the performance period. 7.3.3.3 Forecast Horizon and Lead Time Three-step and six-step ahead forecasts are calculated in addition to one-step ahead forecasts. The lead times considered are also three and six periods. The number of replications is set to 1000. However, for the INARMA(0,0) model with very small mean (??1) more replications are used to reduce the sampling M.Mohammadipour, 2009, Chapter 7 144 error. Therefore, the number of replications for ?= 0.3, 0.5 is 30,000 and for ?= 0.7, 1 is 10,000. 7.3.3.4 Benchmark Methods??Parameters As discussed in chapter 2, three methods of forecasting intermittent demand are selected to compete against the INARMA method. These methods are: Croston (Croston, 1972), SBA (Syntetos and Boylan, 2005) and SBJ (Shale et al., 2006). All of these methods are based on separate smoothing of demand sizes and on the interval between positive demands using a common smoothing parameter for size and interval. Therefore a smoothing constant needs to be selected. It has been suggested in the literature (e.g. Brown, 1959; Croston, 1972) that, especially when the length of history is short, it is best to use fixed values of the smoothing parameter. We choose two arbitrary values for smoothing parameter: ?= 0.2 and ?= 0.5. The first value is selected because in intermittent demand context low smoothing constant values are suggested (Syntetos and Boylan, 2005). However, as can be seen from Table 7-3, some generated series have high autocorrelation; therefore ?= 0.5 is also used. For initialization of the methods, the first inter-demand interval is used as the first smoothed inter-demand interval. For the first smoothed size, the average of the first two positive demands is used. If fewer than two positive demands is observed in the estimation period, the estimation period for that particular replication is extended until two non-zero demands are observed. 7.3.4 Identification Procedure As argued in chapter 4, the sample autocorrelation function (SACF) and partial autocorrelation function (SPACF) of INARMA models have the same structure as those of ARMA models and therefore can be used in identifying the moving average M.Mohammadipour, 2009, Chapter 7 145 and autoregressive orders of the model. However, as argued in section 4.6, for simulation purposes automated methods such as penalty functions should be used. Jung and Tremayne (2003) argue that the first step in analysing time series of counts is to investigate if the data exhibit any serial dependence. If such dependence does not exist, standard methods for independent data should be used. Based on this argument, two identification procedures were suggested in chapter 4, namely, two- stage and one-stage methods. In the two-stage identification method, a Ljung-Box test of Equation 4-6 is first used to test if data has serial dependence. The reasons for the selection of this test were discussed in section 4.6. The second step involves using the AIC of Equation 4-28 (or where applicable, AICC of Equation 4-29) to select the appropriate model among the three possible INARMA models (see section 4.6 for discussion on the application of AIC of ARMA models for INARMA series). In the one-stage identification method, the first step of the previous method is ignored. This means that the AIC is used to select among all possible INARMA models (INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1)). The results of these two methods will be compared in terms of the percentage of series for which the model is identified correctly. This can be done in simulation because the correct model from which the series is produced is known. Another aspect that can be tested is the accuracy of forecasts obtained from each identification method (the accuracy measures are reviewed in section 7.3.7). No identification method can guarantee that the correct model is identified at all times. In such cases, the effect of misidentification on the accuracy of forecasts is of interest. This will also be tested in the next chapter. 7.3.5 Estimation of Parameters As discussed in chapter 5, Yule-Walker (YW) and conditional least squares (CLS) methods are used for estimation of parameters of INAR(1), INMA(1), and INARMA(1,1) processes. Because the conditional maximum likelihood (CML) M.Mohammadipour, 2009, Chapter 7 146 estimation has been established only for INAR(p) processes, we can use it only for an INAR(1) process. For an INARMA(0,0) process, the three estimation methods result in the same estimator. All of these estimators are given in chapter 5. The performance of these estimators has been tested in the literature (Al-Osh and Alzaid, 1987; Br?nn?s, 1994; Bu, 2006). However, this has been done for sample sizes greater than 50. Because we also use smaller numbers of observations, we compare the performance of these estimators. Since the true values of the parameters are known in simulation, we compare the bias and the MSE of the estimates. The impact of estimates on forecast accuracy is also an important issue that has not been looked at before and is covered in this thesis. 7.3.6 Forecasting Method This thesis focuses on comparing the accuracy of forecasts produced by INARMA and benchmark methods. The accuracy measures include MSE and MASE (see section 7.3.7). We use the conditional expected value which yields minimum mean square error (MMSE) forecasts. It has been argued in the literature that this method is not coherent in that it does not produce integer-valued forecasts (Freeland and McCabe, 2004b). Other methods such as conditional median, Markov Chains, and bootstrapping have been suggested to tackle this problem (Cardinal et al., 1999; Freeland and McCabe, 2004b; Jung and Tremayne, 2006b; Bu and McCabe, 2008). However, none of these methods produces MMSE forecasts. Also, those methods that produce the distribution forecast instead of point forecasts are not used for our comparison. Such methods are definitely useful for competing against bootstrap methods for intermittent demand forecasting such as Willemain?s bootstrap (Willemain et al., 2004) and can be considered as a future line of study. The ?-step ahead forecasts and lead time forecasts for INARMA models are discussed in chapter 6 in detail. The Croston, SBA and SBJ forecasts are given in chapter 2. For these methods, the ?-step ahead forecasts are the same as the one-step ahead forecasts and the lead time forecast is simply the one-step ahead forecast multiplied by the length of lead time. M.Mohammadipour, 2009, Chapter 7 147 Finally, two cases regarding the forecast timing are considered: all points in time or focusing on those periods immediately after a positive demand is occurred (issue points). This is because Croston?s method is designed to outperform the SES for issue points and it is of interest to test the performance of the INARMA method for issue points. 7.3.7 Performance Metrics In this section, the performance measures to be used in the simulation are reviewed. In the identification stage, where we want to examine the capability of the two identification procedures, the percentage of correctly identified models is calculated. The accuracy of the forecasts produced by each identification method is also compared. In order to compare the estimation methods (YW, CLS and CML only for INAR(1)), the bias (using Mean Error) and Mean Square Error (MSE) of parameters? estimates are calculated. The performance of the estimates is also compared in terms of their impact on forecast accuracy. Finally, selecting the appropriate forecasting accuracy measure is an important issue for intermittent processes. As discussed in section 2.4, the fact that intermittent demand series include zeros, makes some of the conventional measures inappropriate. The following accuracy measures are used in this thesis: Mean Error (ME), Mean Square Error (MSE), Mean Absolute Scaled Error (MASE) for simulation, along with Percentage Better (PB) of MASE and Relative Geometric Root-Mean-Square Error (RGRMSE) for empirical analysis (see section 2.4.3 for more details). 7.4 Verification Verification is the process to make sure that no programming error has been made (Kleijnen and Groenendaal, 1992). This can be done by calculating some intermediate results manually and comparing them with the results obtained by the program. This M.Mohammadipour, 2009, Chapter 7 148 is called tracing (Kleijnen and Groenendaal, 1992). Eyeballing or reading through the code and looking for bugs is another way of verification (Kleijnen and Groenendaal, 1992). The following steps have been done in order to verify the simulation model: ? The MATLAB code has been read through to make sure that the correct logic and functions have been used. ? The intermediate and also the final results have been compared for a limited number of replications (e.g. 20 replications) with MS Excel. ? The average and standard deviation of the generated INARMA series is calculated and compared to the theoretical mean and standard deviation of the process to test the generated data. The selection of parameters was made to make sure that both highly-intermittent and less-intermittent data are considered. Inter-arrival times are also obtained for each time series. 7.5 Conclusions In this chapter, a simulation experiment was developed to assess the accuracy of approximations made for the mathematical analysis, to measure the accuracy of estimates, to assess the sensitivity of forecast accuracy to control parameters, and to compare the INARMA forecasts with those of benchmark methods. Four integer autoregressive moving average models have been selected for the purpose of simulation (models with ?,??1). The marginal distribution is assumed to be Poisson. The control parameters used are: autoregressive and moving average parameters, innovation parameter, forecast horizon, the length of lead time, and the smoothing parameter for the benchmark methods. As previously discussed in section 2.4.3, different accuracy measures are needed to assess the accuracy of estimates and forecasts. The accuracy of estimates is measured using ME and MSE. Demand being intermittent makes some forecast accuracy measures not applicable. We have selected ME, MSE, and MASE for simulation. The PB of MASE and RGRMSE will be added to the above measures for empirical analysis. M.Mohammadipour, 2009, Chapter 8 149 Chapter 8 SIMULATION RESULTS 8.1 Introduction The simulation results are presented in this chapter. As discussed in chapter 7, the main objective of simulation is to test whether using an INARMA model results in better forecasts compared to benchmark methods of Croston, Syntetos-Boylan Approximation (SBA) and Shale-Boylan-Johnston (SBJ). This is discussed in section 8.6. The simulation experiment also enables us to test the applicability of the Croston-SBA categorization (Syntetos et al., 2005) when demand is an INARMA process. As discussed in section 7.3.1, four processes are simulated: INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1). Based on the arguments in section 2.4, the ME, MSE and MASE of the forecasts are compared to those of Croston, SBA and M.Mohammadipour, 2009, Chapter 8 150 SBJ methods. A range of INARMA parameters and different lengths of history are used (see section 7.3.3). The estimation methods used in this study are YW and CLS for INAR(1), INMA(1) and INARMA(1,1) processes and CML for INAR(1) (see Chapter 5 for detailed discussion). As another objective of simulation, the accuracy of parameters? estimates needs to be tested. The performance of the estimators can be tested not only by comparing the accuracy of the estimates, but also by comparing their impact on the forecast accuracy. The former has been undertaken by comparing the ME and MSE of the parameters? estimates (see section 8.3 and Appendix 8.A). The latter, the results of which are presented in section 8.4, has been accomplished by comparing the ME, MSE and MASE of forecasts obtained using each estimation method. The chapter is structured as follows. Details of the simulation design are reviewed in section 8.2. Sections 8.3 and 8.4 compare the accuracy of different estimates of the parameters of INARMA processes. The Croston-SBA categorization (Syntetos et al., 2005) for data produced by INARMA models is validated in section 8.5. The INARMA forecasts are then compared to the benchmark methods in section 8.6. It is first assumed that the order of the INARMA model is known. The results for the case where the order needs to be identified are presented in section 8.6.2. The lead-time forecasts are compared in section 8.6.3 and the conclusions are provided in section 8.7. 8.2 Details of Simulation As mentioned in chapter 7, the number of replications is set to 1000. However, in order to reduce the sampling error for the case of INARMA(0,0) process with small parameters, (?= 0.3, 0.5) and (?= 0.7, 1), 30,000 and 10,000 replications are used, respectively. It has been suggested in the literature that, especially with short length of history, it is best to use fixed values of smoothing parameters (Brown, 1959; Croston, 1972). Because with intermittent demand, data history is short in most cases, we use two arbitrary values for the smoothing parameter for Croston, SBA and SBJ (?= 0.2 and M.Mohammadipour, 2009, Chapter 8 151 ?= 0.5). As summarized in chapter 7, the initialization for Croston, SBA and SBJ is based on using the first inter-demand interval as the first smoothed inter-demand interval and the average of the first two non-zero observations as the first smoothed size. The observations are divided into two categories: estimation period and performance period. Initialization and estimation of parameters are conducted in the estimation period and the estimates? accuracy and forecasting accuracy are assessed in the performance period. If at least two non-zero demands are observed in the estimation period, the first half of the observations is assigned for the estimation period and the other half for the performance period. However, if fewer than two non-zero demands are observed in the estimation period, this period will be extended until the second non-zero demand is observed. In order to obtain a stationary series, we initialize the INARMA methods with the expected value of each model. As discussed in chapter 7, the forecasting accuracy is obtained for both cases of all points in time and issue points (i.e. after a positive demand is observed). Finally, if there is no nonzero observation in the performance period, the error measures for issue points are excluded (only for the corresponding replication). If the in-sample MAE is zero, the MASE for that replication is excluded. 8.3 Accuracy of INARMA Parameter Estimates As previously discussed in chapter 5, two methods (YW and CLS) have been used to estimate the parameters of all four INARMA processes. In this section, the accuracy of these parameter estimates is evaluated using MSE. Out of the four INARMA processes of this study, only three are included for comparison of estimation methods: INAR(1), INMA(1) and INARMA(1,1). The YW, CLS and CML estimates for INARMA(0,0) are the same (see section 5.2). As previously mentioned, CML is also used in addition to YW and CLS in order to estimate the parameters of an INAR(1) process. The reason for excluding CML for other processes is that the maximum likelihood functions for INMA(1) and M.Mohammadipour, 2009, Chapter 8 152 INARMA(1,1) processes have not been developed in the literature (see Chapter 5). The parameters may fall out of the region [0,1]. In order to tackle this issue, the parameters are set equal to their closest boundary value in each case (Br?nn?s and Hall, 2001). The accuracy of YW, CLS and CML estimates of the parameters of an INAR(1) process for the case of ?= 24 are compared in Table ?8-1. For high values of ? and also when the mean of the process, ?/(1??), is high, the CML becomes computationally expensive. Table 8-1 MSE of YW, CLS and CML estimates for INAR(1) series when Parameters ? ? YW CLS CML YW CLS CML ?= 0.1,?= 0.5 0.0203 0.0238 0.0337 0.0391 0.0394 0.0447 ?= 0.5,?= 0.5 0.0772 0.0762 0.0734 0.1381 0.1337 0.1043 ?= 0.9,?= 0.5 0.1436 0.1056 0.0042 3.9057 2.8683 0.0891 ?= 0.1,?= 1 0.0196 0.0226 0.0364 0.0877 0.0910 0.1104 ?= 0.5,?= 1 0.0723 0.0704 0.0631 0.4136 0.3950 0.3125 ?= 0.9,?= 1 0.1429 0.1067 0.0028 14.7487 11.0440 0.2526 ?= 0.1,?= 3 0.0188 0.0215 0.0419 0.4225 0.4509 0.6832 ?= 0.5,?= 3 0.0716 0.0684 0.0658 2.9762 2.8352 2.6277 ?= 0.9,?= 3 0.1462 0.1124 0.0024 134.0715 102.9696 2.0292 ?= 0.1,?= 5 0.0197 0.0227 0.0390 0.9650 1.0462 1.6005 ?= 0.5,?= 5 0.0710 0.0686 0.0606 7.6814 7.4755 6.4135 The results confirm that, as suggested by Al-Osh and Alzaid (1987), the MSE of estimates produced by CML is generally less than that of YW and CLS (with the exception of the cases where ?= 0.1). However, it will be seen in a later section that the results of CML in terms of its effect on forecast accuracy are not very far from those by YW and CLS. This is also true for those cases in Table ?8-1 that the MSE of CML is much less than that of the other methods (e.g. ?= 0.9 and ?= 3). The results of comparing the MSE of YW and CLS estimates of the parameters of INAR(1), INMA(1) and INARMA(1,1) processes are shown in Table ?8-2, Table ?8-3, and Table ?8-4, respectively. Al-Osh and Alzaid (1987) suggest that the accuracy of YW and CLS estimates for parameters of an INAR(1) process are close. The results of Table ?8-2 confirm this 24?n M.Mohammadipour, 2009, Chapter 8 153 when the number of observations is high. However, for fewer observations, the difference is high when the autoregressive parameter is high. Table 8-2 Accuracy of YW and CLS estimates for INAR(1) series Parameters MSE(???)/MSE(????) MSE(???)/MSE(????) ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 0.1,?= 0.5 0.8402 0.8978 0.9351 0.9700 1.0000 0.9868 0.9918 0.9890 1.0000 1.0000 ?= 0.5,?= 0.5 1.0291 1.0291 1.0199 1.0248 1.0000 1.0408 1.0289 1.0194 1.0182 1.0000 ?= 0.9,?= 0.5 1.3036 1.3487 1.3063 1.2404 1.1250 1.3089 1.3355 1.2935 1.2440 1.0874 ?= 0.1,?= 1 0.8522 0.9222 0.9388 0.9691 1.0000 0.9655 0.9728 0.9762 0.9891 1.0000 ?= 0.5,?= 1 1.0255 1.0227 1.0278 1.0286 1.0000 1.0213 1.0348 1.0227 1.0230 1.0000 ?= 0.9,?= 1 1.3502 1.2990 1.3029 1.2294 1.1429 1.3550 1.3101 1.3004 1.2291 1.0957 ?= 0.1,?= 3 0.8610 0.9102 0.9379 0.9655 1.0000 0.9228 0.9422 0.9644 0.9806 0.9946 ?= 0.5,?= 3 1.0319 1.0349 1.0224 1.0148 1.0000 1.0317 1.0336 1.0307 1.0185 1.0024 ?= 0.9,?= 3 1.3285 1.3063 1.3038 1.2526 1.1429 1.3273 1.3012 1.3010 1.2485 1.0954 ?= 0.1,?= 5 0.8649 0.9118 0.9416 0.9670 1.0000 0.9094 0.9370 0.9565 0.9776 0.9959 ?= 0.5,?= 5 1.0290 1.0396 1.0224 1.0216 1.0000 1.0355 1.0400 1.0240 1.0204 1.0068 ?= 0.9,?= 5 1.3288 1.2945 1.3123 1.2376 1.1429 1.3236 1.2904 1.3110 1.2391 1.0960 The results show that for an INAR(1) process, when the number of observations is small, for high values of ?, CLS produces much better estimates for both ? and ? in terms of MSE (up to 35 percent improvement in MSE). On the other hand, for small values of ?, YW results in better estimates (up to 16 percent improvement in MSE). The results of section 8.4 show that this is also true for the accuracy of forecasts produced by these estimates. The MSE of ? for both YW and CLS estimates increases with an increase in ? but this is not necessarily the case for the MSE of ? (see Appendix 8.A). This confirms the argument by Al-Osh and Alzaid (1987). The results of Table ?8-3 show that for an INMA(1) series, for a small number of observations, CLS has smaller MSE than YW except for the case of ?= 0.9. When the number of observations increases, for high values of ?, the MSE of YW estimates decreases with a greater pace compared to CLS. However, as will be discussed in section 8.4, it does not have a great effect on the accuracy of forecasts produced by each method. The MSE of ? for both YW and CLS estimates increases with an increase in ? but the same is not necessarily true for the MSE of ? (see Appendix 8.A). M.Mohammadipour, 2009, Chapter 8 154 Table 8-3 Accuracy of YW and CLS estimates for INMA(1) series Parameters MSE(???)/MSE(????) MSE(???)/MSE(????) ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 0.1,?= 0.5 2.0558 1.9491 2.0409 1.9897 1.3929 0.9838 1.0084 1.0265 1.0510 1.0500 ?= 0.5,?= 0.5 1.1761 1.2907 1.1692 1.1307 0.5213 0.9008 0.9189 0.9032 0.8563 0.6098 ?= 0.9,?= 0.5 0.8054 0.6843 0.6217 0.5340 0.2336 0.9462 0.8521 0.8112 0.7927 0.5000 ?= 0.1,?= 1 1.9449 2.0335 2.0395 1.7292 1.3793 0.9952 1.0249 1.0518 1.0909 1.0962 ?= 0.5,?= 1 1.1879 1.1665 1.2214 1.1247 0.5354 0.8940 0.9045 0.8977 0.8772 0.5968 ?= 0.9,?= 1 0.7304 0.6812 0.5904 0.4636 0.2263 0.8650 0.8296 0.8080 0.7352 0.4012 ?= 0.1,?= 3 2.2829 2.1860 2.1367 1.8646 1.4615 1.0799 1.1157 1.1649 1.2067 1.1815 ?= 0.5,?= 3 1.1216 1.1679 1.1264 1.0872 0.4928 0.8906 0.9078 0.9011 0.8400 0.5050 ?= 0.9,?= 3 0.6900 0.6048 0.5664 0.3868 0.1909 0.8005 0.7554 0.7359 0.6084 0.3300 ?= 0.1,?= 5 2.4159 2.4242 2.2715 1.8780 1.4074 1.1174 1.2171 1.2425 1.2495 1.1940 ?= 0.5,?= 5 1.0589 1.0776 1.1294 1.0726 0.4634 0.8654 0.8406 0.8873 0.8217 0.4519 ?= 0.9,?= 5 0.6060 0.5175 0.4474 0.3523 0.1727 0.7006 0.6274 0.5856 0.5120 0.2597 The results of Table ?8-4 show that, for INARMA(1,1) series, CLS produces better estimates especially when the number of observations is small and the autoregressive parameter is high. This is also true for the accuracy of forecasts produced by CLS compared to those by YW (as shown later). To conclude, for INAR(1), INMA(1), and INARMA(1,1) processes, the autoregressive and moving average parameters and the number of observations determine which estimation method produces more accurate estimates. For an INAR(1) process, CLS outperforms YW for high values of ?. The same is generally true for an INMA(1) process with low values of ? and small number of observations. Finally, for an INARMA(1,1) process, CLS generally produces better estimates than YW with a few exceptions. 8.4 Forecasting Accuracy of INARMA Estimation Methods As previously discussed in chapter 5, two methods (CLS and YW) have been used to estimate the parameters of all four INARMA processes. In this section, the accuracy of these estimates in terms of their effect on forecast accuracy is evaluated. The forecast accuracy is measured by ME, MSE, and MASE (see section 2.4 for detailed discussion). We focus on MSE in this section. MSE is specially selected because of its theoretical tractability. Also due to the fact that data is theoretically generated, the scale- dependency problem is not an issue when we average across multiple series. M.Mohammadipour, 2009, Chapter 8 155 Table 8-4 Accuracy of YW and CLS estimates for INARMA(1,1) series Parameters MSE(???)/MSE(????) MSE(???)/MSE(????) MSE(???)/MSE(????) ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 0.1,?= 0.1,?= 0.5 4.8614 5.4738 6.0749 7.0578 3.4853 1.7669 1.9854 1.9831 1.9153 1.6269 2.0720 2.6889 3.1239 3.9265 3.7576 ?= 0.1,?= 0.9,?= 0.5 0.8613 0.7344 0.6736 0.6839 0.9770 0.6373 0.5349 0.4602 0.3182 0.1052 1.0420 0.9860 0.9155 0.7370 0.3245 ?= 0.5,?= 0.5,?= 0.5 1.3778 1.5195 1.5427 1.4870 1.1607 1.2698 1.2589 1.3484 1.3281 0.5957 1.0321 0.9228 0.8333 0.6433 0.2514 ?= 0.9,?= 0.1,?= 0.5 2.1067 2.0306 1.8965 1.5271 1.2222 26.3171 27.4400 28.9136 20.4444 3.8571 1.2033 1.1088 1.0270 0.9594 0.8696 ?= 0.1,?= 0.1,?= 1 4.9796 6.3395 6.5819 5.9756 3.6176 2.0247 1.8398 1.7630 1.7983 1.5588 2.7496 3.5359 3.9433 4.7348 4.6538 ?= 0.1,?= 0.9,?= 1 0.8243 0.5925 0.6154 0.6291 0.8710 0.5975 0.4733 0.4273 0.2969 0.1013 1.0431 0.9933 0.9032 0.7263 0.2915 ?= 0.5,?= 0.5,?= 1 1.6988 1.7842 1.7343 1.6031 1.2157 1.0136 1.0992 1.1177 1.0436 0.5140 1.0337 0.9391 0.8374 0.6111 0.2446 ?= 0.9,?= 0.1,?= 1 2.3724 2.2172 2.0000 1.7658 1.3750 43.0375 35.2133 31.3239 19.9841 2.8772 1.1029 1.0416 0.9534 0.9135 0.8853 ?= 0.1,?= 0.1,?= 5 5.2880 6.5479 7.3636 8.0300 3.3701 2.5138 2.3421 2.1606 1.9722 1.6857 4.4277 5.6560 6.6359 8.6472 5.6930 ?= 0.1,?= 0.9,?= 5 0.5406 0.4332 0.3695 0.3471 0.4227 0.4187 0.3415 0.2803 0.1872 0.0582 1.1328 1.0459 0.9843 0.7719 0.2663 ?= 0.5,?= 0.5,?= 5 2.2217 2.3401 2.5780 2.1421 1.0517 0.8303 0.8293 0.8479 0.7495 0.4093 0.9961 0.9877 0.9281 0.7039 0.2679 ?= 0.9,?= 0.1,?= 5 2.5276 2.3845 2.2099 1.8218 1.5000 45.5789 37.9867 31.5135 17.3108 2.4211 1.0838 0.9427 0.8756 0.8296 0.8635 M.Mohammadipour, 2009, Chapter 8 156 The effect of CLS and YW estimates on one-step ahead forecasts is presented in Table 8-5, Table 8-6, and Table 8-7. The results for three-step ahead and six-step ahead forecasts are presented in Table 8-8 to Table ?8-13. For an INAR(1) process, the results of YW and CLS are compared to those of CML for ?= 24. For longer length of history, the CML results become computationaly expensive. The results for an INAR(1) process show that when the history is short and data is highly autocorrelated (?= 0.9), CLS produces more accurate forecasts (up to 11 percent improvement in MSE) than YW. For ??0.5, YW produces better forecasts, but the magnitude of improvement is small (up to a maximum of 3 percent improvement in MSE). The results also confirm that when the number of observations increases, the two methods yield very similar forecast errors (Al-Osh and Alzaid, 1987; Bu, 2006). Table 8-5 One-step ahead forecast error comparison (YW, CLS and CML) for INAR(1) series Parameters MSEYW /MSECLS MSEYW /MSECML ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 0.1,?= 0.5 0.9752 0.9841 0.9936 0.9978 1.0000 0.9727 ?= 0.5,?= 0.5 0.9847 0.9954 0.9974 0.9999 1.0001 0.9799 ?= 0.9,?= 0.5 1.1087 1.0957 1.0560 1.0248 1.0010 1.2084 ?= 0.1,?= 1 0.9774 0.9926 0.9956 0.9985 0.9999 0.9751 ?= 0.5,?= 1 0.9871 1.0006 0.9984 0.9995 1.0001 0.9899 ?= 0.9,?= 1 1.1268 1.0836 1.0637 1.0249 1.0012 1.2329 ?= 0.1,?= 3 0.9842 0.9935 0.9962 0.9987 0.9999 0.9694 ?= 0.5,?= 3 0.9877 1.0009 1.0019 1.0000 1.0001 0.9850 ?= 0.9,?= 3 1.0993 1.0807 1.0568 1.0228 1.0009 1.2605 ?= 0.1,?= 5 0.9849 0.9922 0.9960 0.9987 0.9999 0.9687 ?= 0.5,?= 5 0.9925 1.0000 1.0004 1.0003 1.0001 1.0019 ?= 0.9,?= 5 1.1133 1.0866 1.0647 1.0252 1.0011 - As noted previously, it is computationally expensive to calculate CML when the mean of the process is high. Therefore, no result is presented for the last case in Table ?8-5. The results also show that, except for the case where the autoregressive parameter is high, YW forecasts have smaller MSE than CML forecasts for ?= 24. The above discussion about YW and CLS suggests that for such cases CLS is better than YW, but the results show that CML is still better than CLS for these cases. The results for an INMA(1) process show that the forecast accuracy of YW and CLS estimates are generally close. When the history is short, CLS produces better M.Mohammadipour, 2009, Chapter 8 157 forecasts for lower values of ? (up to 1.4 percent improvement in MSE). As shown in Table ?8-6, for high values of ?, YW outperforms CLS (up to 3 percent improvement in MSE). Table 8-6 One-step ahead forecast error comparison (YW and CLS) for INMA(1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 0.1,?= 0.5 1.0000 1.0003 1.0008 1.0004 1.0000 ?= 0.5,?= 0.5 1.0065 1.0080 1.0054 1.0034 1.0008 ?= 0.9,?= 0.5 1.0137 1.0156 1.0111 1.0069 1.0013 ?= 0.1,?= 1 0.9998 1.0009 1.0005 1.0004 1.0001 ?= 0.5,?= 1 1.0058 1.0040 1.0045 1.0037 1.0031 ?= 0.9,?= 1 1.0084 1.0062 1.0075 1.0019 1.0003 ?= 0.1,?= 3 0.9985 0.9999 1.0003 1.0004 1.0000 ?= 0.5,?= 3 1.0020 1.0032 1.0006 1.0019 0.9998 ?= 0.9,?= 3 0.9930 0.9888 0.9915 0.9914 0.9944 ?= 0.1,?= 5 0.9954 0.9986 0.9992 0.9999 0.9999 ?= 0.5,?= 5 0.9972 0.9957 0.9964 0.9990 0.9989 ?= 0.9,?= 5 0.9888 0.9850 0.9857 0.9835 0.9885 For an INARMA(1,1) process the results show that CLS always produces better forecasts than YW. As shown in Table ?8-7, when ??0.5, CLS outperforms YW by up to 20 percent. The difference is much greater when ?= 0.9 (up to 90 percent improvement in MSE). However, with an increase in the number of observations, the two methods become closer, especially for ?= 0.9. Table 8-7 One-step ahead forecast error comparison (YW and CLS) for INARMA(1,1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 0.1,?= 0.1,?= 0.5 1.0850 1.1076 1.1175 1.0991 1.0259 ?= 0.1,?= 0.9,?= 0.5 1.0273 1.0556 1.0551 1.0499 1.0185 ?= 0.5,?= 0.5,?= 0.5 1.0633 1.0842 1.0805 1.0467 1.0087 ?= 0.9,?= 0.1,?= 0.5 1.4006 1.2431 1.1654 1.0475 1.0032 ?= 0.1,?= 0.1,?= 1 1.0992 1.1077 1.1015 1.0874 1.0252 ?= 0.1,?= 0.9,?= 1 1.0527 1.0624 1.0700 1.0568 1.0188 ?= 0.5,?= 0.5,?= 1 1.1293 1.1172 1.1031 1.0565 1.0104 ?= 0.9,?= 0.1,?= 1 1.4797 1.2853 1.1771 1.0561 1.0031 ?= 0.1,?= 0.1,?= 5 1.1110 1.1142 1.1175 1.1074 1.0246 ?= 0.1,?= 0.9,?= 5 1.1254 1.1285 1.1397 1.1273 1.0693 ?= 0.5,?= 0.5,?= 5 1.1939 1.1694 1.1603 1.0789 1.0232 ?= 0.9,?= 0.1,?= 5 1.9098 1.4115 1.2442 1.0676 1.0042 Based on the above results for one-step ahead forecasts, for INMA(1), YW and CLS M.Mohammadipour, 2009, Chapter 8 158 are close. For INAR(1), when the history is short and the autoregressive parameter is high, CLS is considerably better than YW. But when ??0.5, the difference is much smaller. For INARMA(1,1) and especially for short history, CLS estimates produce better results than YW. Although the above results are based on MSE, using MASE produces similar results (see Appendix 8.B). For INAR(1), CLS produces more accurate forecasts (up to 9 percent improvement in MASE) when the history is short and the autoregressive parameter is high (?= 0.9). For ??0.5, YW produces better forecasts, but the magnitude of improvement is small (up to a maximum of 3 percent improvement in MASE). For an INMA(1) process, the forecasting accuracy of YW and CLS forecasts using MASE are very close. For an INARMA(1,1) process, CLS produces better forecasts than YW in most of the cases (up to 30 percent improvement in MASE). Finally, the results confirm that when the number of observations increases, the two methods become very close in terms of MASE. The results for three-step and six-step ahead forecasts for INAR(1) process are shown in Table ?8-8 and Table ?8-9. Although the three-step ahead results follow the same pattern as one-step ahead forecasts, both three-step and six-step ahead forecasts produced by YW and CLS estimates are very close. The results of YW and CLS three-step and six-step ahead forecasts for INMA(1) process are presented in Table ?8-10 and Table ?8-11. It can be seen that the two estimation methods result in very close forecasts. Table 8-8 Three-step ahead forecast error comparison (YW and CLS) for INAR(1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 0.1,?= 0.5 0.9988 0.9992 0.9997 0.9998 ?= 0.5,?= 0.5 0.9906 0.9993 1.0009 1.0013 ?= 0.9,?= 0.5 1.0081 1.0189 1.0229 1.0382 ?= 0.1,?= 1 0.9989 0.9996 0.9996 1.0000 ?= 0.5,?= 1 0.9965 0.9995 1.0001 1.0012 ?= 0.9,?= 1 1.0181 1.0210 1.0170 1.0367 ?= 0.1,?= 3 0.9982 0.9994 0.9998 0.9999 ?= 0.5,?= 3 0.9933 0.9995 1.0003 1.0014 ?= 0.9,?= 3 1.0070 1.0135 1.0245 1.0362 ?= 0.1,?= 5 0.9990 0.9998 0.9999 0.9999 ?= 0.5,?= 5 0.9938 0.9997 1.0005 1.0014 ?= 0.9,?= 5 1.0033 1.0157 1.0118 1.0352 M.Mohammadipour, 2009, Chapter 8 159 Table 8-9 Six-step ahead forecast error comparison (YW and CLS) for INAR(1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 0.1,?= 0.5 0.9986 0.9995 0.9998 1.0000 ?= 0.5,?= 0.5 0.9848 0.9938 0.9979 0.9993 ?= 0.9,?= 0.5 0.9701 0.9786 0.9936 1.0139 ?= 0.1,?= 1 0.9989 0.9997 0.9997 0.9999 ?= 0.5,?= 1 0.9833 0.9936 0.9969 0.9994 ?= 0.9,?= 1 0.9509 0.9723 0.9897 1.0138 ?= 0.1,?= 3 0.9987 0.9995 0.9998 0.9999 ?= 0.5,?= 3 0.9907 0.9918 0.9980 0.9994 ?= 0.9,?= 3 0.9576 0.9679 0.9880 1.0175 ?= 0.1,?= 5 0.9984 0.9996 0.9996 0.9999 ?= 0.5,?= 5 0.9883 0.9952 0.9960 0.9992 ?= 0.9,?= 5 0.9566 0.9738 0.9854 1.0101 Table 8-10 Three-step ahead forecast error comparison (YW and CLS) for INMA(1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 0.1,?= 0.5 0.9940 0.9957 0.9980 0.9988 ?= 0.5,?= 0.5 0.9895 0.9909 0.9939 0.9967 ?= 0.9,?= 0.5 0.9811 0.9886 0.9918 0.9959 ?= 0.1,?= 1 0.9948 0.9964 0.9988 0.9984 ?= 0.5,?= 1 0.9934 0.9916 0.9929 0.9968 ?= 0.9,?= 1 0.9774 0.9851 0.9896 0.9954 ?= 0.1,?= 3 0.9947 0.9961 0.9968 0.9981 ?= 0.5,?= 3 0.9878 0.9940 0.9953 0.9963 ?= 0.9,?= 3 0.9951 0.9917 0.9932 0.9953 ?= 0.1,?= 5 0.9927 0.9957 0.9976 0.9985 ?= 0.5,?= 5 0.9940 0.9935 0.9948 0.9988 ?= 0.9,?= 5 0.9970 0.9998 0.9970 0.9978 Table 8-11 Six-step ahead forecast error comparison (YW and CLS) for INMA(1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 0.1,?= 0.5 0.9927 0.9955 0.9970 0.9984 ?= 0.5,?= 0.5 0.9846 0.9946 0.9925 0.9973 ?= 0.9,?= 0.5 0.9798 0.9953 0.9942 0.9960 ?= 0.1,?= 1 0.9921 0.9970 0.9975 0.9989 ?= 0.5,?= 1 0.9907 0.9908 0.9939 0.9955 ?= 0.9,?= 1 0.9821 0.9844 0.9909 0.9945 ?= 0.1,?= 3 0.9929 0.9931 0.9968 0.9989 ?= 0.5,?= 3 0.9917 0.9911 0.9942 0.9963 ?= 0.9,?= 3 0.9919 0.9936 0.9953 0.9934 ?= 0.1,?= 5 0.9935 0.9954 0.9969 0.9980 ?= 0.5,?= 5 0.9919 0.9910 0.9967 0.9984 ?= 0.9,?= 5 1.0013 0.9958 0.9935 0.9980 M.Mohammadipour, 2009, Chapter 8 160 For an INARMA(1,1) process, as can be seen from Table ?8-12 and Table 8-13, CLS does not always produce better forecasts than YW. For high number of observations, the results are close. Table 8-12 Three-step ahead forecast error comparison (YW and CLS) for INARMA(1,1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 0.1,?= 0.1,?= 0.5 1.0301 1.0092 1.0051 1.0020 ?= 0.1,?= 0.9,?= 0.5 0.9986 0.9928 0.9977 0.9973 ?= 0.5,?= 0.5,?= 0.5 0.9752 0.9976 0.9965 1.0009 ?= 0.9,?= 0.1,?= 0.5 0.8939 0.9184 0.9467 1.0027 ?= 0.1,?= 0.1,?= 1 1.0217 1.0082 1.0100 1.0022 ?= 0.1,?= 0.9,?= 1 0.9825 0.9949 0.9958 0.9968 ?= 0.5,?= 0.5,?= 1 0.9531 1.0014 1.0013 1.0004 ?= 0.9,?= 0.1,?= 1 0.9139 1.0007 0.9974 1.0353 ?= 0.1,?= 0.1,?= 5 1.0147 1.0143 1.0072 1.0003 ?= 0.1,?= 0.9,?= 5 0.9696 0.9956 0.9892 0.9921 ?= 0.5,?= 0.5,?= 5 0.9635 0.9725 0.9923 0.9919 ?= 0.9,?= 0.1,?= 5 1.1665 1.0307 1.0383 1.0639 Table 8-13 Six-step ahead forecast error comparison (YW and CLS) for INARMA(1,1) series Parameters MSEYW /MSECLS ?= 24 ?= 36 ?= 48 ?= 96 ?= 0.1,?= 0.1,?= 0.5 1.0094 1.0088 1.0050 1.0006 ?= 0.1,?= 0.9,?= 0.5 0.9936 0.9933 0.9977 0.9975 ?= 0.5,?= 0.5,?= 0.5 0.9720 0.9965 0.9975 0.9976 ?= 0.9,?= 0.1,?= 0.5 0.8577 0.8722 0.9262 0.9797 ?= 0.1,?= 0.1,?= 1 1.0157 1.0127 1.0047 1.0010 ?= 0.1,?= 0.9,?= 1 0.9735 0.9956 0.9964 0.9963 ?= 0.5,?= 0.5,?= 1 0.9659 0.9947 0.9939 0.9959 ?= 0.9,?= 0.1,?= 1 0.9251 0.9431 0.9648 0.9977 ?= 0.1,?= 0.1,?= 5 1.0072 1.0193 1.0027 1.0017 ?= 0.1,?= 0.9,?= 5 0.9688 0.9872 0.9930 0.9919 ?= 0.5,?= 0.5,?= 5 0.9167 0.9718 0.9769 0.9907 ?= 0.9,?= 0.1,?= 5 1.1674 1.0402 1.0173 1.0190 For an INAR(1) process, the results of this section show that CLS produces better one-step ahead forecasts than YW for high autoregressive parameters. For lower autoregressive parameters, YW slightly outperform CLS. For an INMA(1) process, generally for high values of ?, YW is slightly better than CLS in terms of MSE of one-step ahead forecasts; but the opposite is true for lower values of ?. For an INARMA(1,1) process, CLS always produces better one-step ahead forecast than M.Mohammadipour, 2009, Chapter 8 161 YW using MSE. However, the results of three-step and six-step ahead forecasts show that although the YW and CLS results are generally very close for all of the three INARMA processes, YW results are slightly better in many cases. Therefore, based on the superior performance of CLS for one-step ahead forecasts, in the following sections where we compare the INARMA forecasts with those of benchmark methods, we use the CLS to estimate the parameters for one-step ahead INARMA forecasts. For three-step and six-step ahead forecasts, on the other hand, we use YW to estimate the parameters of the INARMA process. 8.5 Croston-SBA Categorization Syntetos et al. (2005) compare Croston and SBA, based on MSE, to establish the areas that each method should be used over the other. The squared coefficient of variation (??2) of demand size and the average inter-demand interval (?) are used to identify the areas. The coefficient of variation is defined by ?=?/?1 ?, where ?? is the value of the mean measured from some arbitrary origin. It is estimated using the formula: ?= ? 1 ? ? (????1 ?)2??=1 ? 1/2 1 ? ? ?? ? ?=1 Equation 8-1 where ? is the sample size. The results show that for the smoothing parameter ? = 0.2, when p > 1.31, SBA is superior to Croston?s method in terms of MSE. For p?1.31, if CV2 > 0.47 then MSECroston > MSESBA , but if CV 2 ?0.47 then Croston?s method performs better in terms of MSE (Syntetos et al., 2005). This is shown in Figure 8-1. The cut-off values are slightly different for different smoothing parameters (Syntetos et al., 2005). The Croston-SBA categorization is based on the assumption that demand occurs as an i.i.d. Bernoulli process. Therefore it is worth testing if it also holds for an i.i.d. M.Mohammadipour, 2009, Chapter 8 162 Poisson process. An INARMA(0,0) process produces such data series. Figure ?8-1 Cut-off values for Croston and SBA when (Syntetos et al., 2005) The simulation results show that the squared coefficient of variation of demand size (??2) is always less than the cut-off value determined by Syntetos et al. (2005) (0.47). This is due to the Poisson assumption of demand. The p-values, however, vary below and beyond the cut-off value (1.31). Therefore, the demand series produced by an INARMA(0,0) process could belong to either region 3 or 4 in Figure 8-1. The results of simulation confirm the Croston-SBA categorization for both cases of ?= 0.2, 0.5. Therefore, for ??1.31 (or ??3), MSECroston < MSESBA and for ?> 1.31 (or ?< 3), MSESBA < MSECroston when either all points in time or issue points are considered. Although the Croston-SBA categorization is based on MSE, the results show that it generally holds for MASE as well. This was expected due to the similarities between the two error measures. However, there are some exceptions. For the case of ?= 3 where Croston?s method should outperform SBA, MASESBA < MASECroston for both cases of ?= 0.2, 0.5. Because the corresponding ?-value and ??2 of size for these cases are ?= 1.0538 and ??2 = 0.2622, these exceptions can be attributed to the nonlinear boundaries between region 3 and others in Figure 8-1 (Syntetos, 2001; Kostenko and Hyndman, 2006). This is an interesting finding because MASE is a relatively new measure and has been suggested for intermittent demand studies 20.?? Syntetos & Boylan 2 Syntetos & Boylan 1 Syntetos & Boylan 4 Croston 3 ?= 0.47 ?= 1.31 M.Mohammadipour, 2009, Chapter 8 163 (Hyndman, 2006). The results show that when the number of observations increases, the advantage of SBA over Croston decreases. This is a new finding and this issue has not been discussed previously in the literature. It is shown is Table 8-14. Table 8-14 The advantage of SBA over Croston for and all points in time Parameters MSESBA ?MSECroston MASESBA ?MASECroston ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 0.3 -0.0186 -0.0085 -0.0051 -0.0025 -0.0025 -0.0507 -0.0431 -0.0392 -0.0355 -0.0342 ?= 0.5 -0.0168 -0.0084 -0.0061 -0.0051 -0.0052 -0.0245 -0.0197 -0.0176 -0.0175 -0.0174 ?= 0.7 -0.0173 -0.0100 -0.0082 -0.0079 -0.0077 -0.0116 -0.0076 -0.0067 -0.0069 -0.0063 ?= 1 -0.0173 -0.0118 -0.0111 -0.0107 -0.0102 -0.0152 -0.0131 -0.0134 -0.0130 -0.0131 Table 8-15 The advantage of SBA over Croston for and all points in time Parameters MSESBA ?MSECroston MASESBA ?MASECroston ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 0.3 -0.0407 -0.0266 -0.0229 -0.0209 -0.0215 -0.1285 -0.1168 -0.1102 -0.1043 -0.1000 ?= 0.5 -0.0495 -0.0430 -0.0403 -0.0407 -0.0390 -0.0733 -0.0673 -0.0629 -0.0624 -0.0612 ?= 0.7 -0.0623 -0.0595 -0.0547 -0.0562 -0.0558 -0.0486 -0.0473 -0.0438 -0.0461 -0.0439 ?= 1 -0.0780 -0.0743 -0.0761 -0.0715 -0.0696 -0.0442 -0.0427 -0.0441 -0.0415 -0.0416 It can be seen from Table ?8-14 that the advantage of SBA over Croston decreases when ? increases. The MSE of one step ahead forecast for a stationary mean model is: MSE = var?Estimates?+ Bias2 + var(Actual Demand) Equation 8-2 Syntetos (2001) assumes an infinite history for SES estimates: ??= ? ?(1??) ????? ? ?=0 Equation 8-3 and, therefore, Brown?s expression for the variance of estimates is independent of the length of demand history, ?: 20.?? 50.?? M.Mohammadipour, 2009, Chapter 8 164 var(??) = ? 2?? var(??) Equation 8-4 However, when this assumption is relaxed, the finite representation of SES becomes (Graves, 1999): ??=????1 +??1??????2 +?+??1??? ??1????+1 +?1??? ????? Equation 8-5 Therefore, the variance of the estimates produced by SES with a finite history is: var????= ?+ 2(1??)2?+1 2?? var???? Equation 8-6 The variances of the exponentially smoothed size of demand and inter-demand interval with finite observations are then: var??? ? = ?+ 2(1??)2?+1 2?? var???? Equation 8-7 var??? ? = ?+ 2(1??)2?+1 2?? var???? Equation 8-8 where var????=? 2 and var????=?(??1), since the inter-demand interval follows the geometric distribution. The variance of the estimates produced by Croston?s method is: var??? ? = var? ?? ? ?? ? Equation 8-9 The variance of the ratio of two independent variables is given by (Stuart and Ord, 1994): var? ? ? ?=? ???? ???? ? 2 ? var(?) [????]2 + var(?) [????]2 ? Equation 8-10 M.Mohammadipour, 2009, Chapter 8 165 The variance of the estimates produced by Croston?s method with finite sample is therefore: var??? ? = ?+ 2(1??)2?+1 2?? ? (??1)2?2 ?4 + ?2 ?2 ? Equation 8-11 It can be seen that for ??(0,1], as ? increases, the above coefficient decreases until it reaches a limit of [?/(2??)]. For high values of ?, the limit is approached very quickly. The difference between MSE of SBA and Croston?s method is: MSESBA ?MSECroston =?var?EstimatesSBA?+ BiasSBA 2 ??[var?EstimatesCroston ?+ BiasCroston 2 ] Equation 8-12 But when ? increases, the bias of both Croston and SBA decreases and is close to zero (this has been confirmed by simulation results). So the difference is approximately: MSESBA ?MSECroston ???1? ? 2 ? 2 ?1?var?EstimatesCroston ? Equation 8-13 From Equation 8-11 we have: MSESBA ?MSECroston ???1? ? 2 ? 2 ?1?? ?+ 2(1??)2?+1 2?? ?? (??1)2?2 ?4 + ?2 ?2 ? Equation 8-14 As the results of Table 8-14 show, when ? = 0.2, the above coefficient decreases when ? increases; therefore the difference between MSE of Croston and SBA also decreases. However, because the above coefficient reaches a limit of ? ?2 4 ???? ? 2?? ?, it can be seen from Table 8-14 that the advantage of SBA over Croston does not change perceptibly when the number of observations is high. For ?= 0.5, the results of Table ?8-15 confirm that, as expected, the difference between MSE of Croston and SBA changes little with changes in ?. M.Mohammadipour, 2009, Chapter 8 166 Although the Croston-SBA categorization is for i.i.d. demand, we have also tested it when demand is an INAR(1), INMA(1) or an INARMA(1,1) process. The results confirm that the Croston-SBA categorization generally holds for all of the above- mentioned processes. There is only one exception when there is an autoregressive component (either INAR(1) or INARMA(1,1)). For the INMA(1) case, there are two exceptions to the Croston-SBA categorization. The results are presented in Appendix 8.C. Therefore, because the Croston-SBA categorization generally holds when the data is produced by any of the four INARMA processes, the best benchmark can be used to compete with INARMA forecasting methods. 8.6 INARMA vs Benchmark Methods This research has suggested using INARMA models to forecast intermittent demand. In order to answer the last research question of ?Do INARMA models provide more accurate forecasts for intermittent demand than non-optimal smoothing-based methods??, the performance of INARMA forecasts based on ME, MSE and MASE has been compared to that of benchmark methods. As previously mentioned, the benchmarks are Croston (Croston, 1972), SBA (Syntetos and Boylan, 2005) and SBJ (Shale et al., 2006) methods. The first steps in the INARMA methodology are identification and estimation. These steps make INARMA more complicated than the benchmarks and result in two types of errors: error of identification and error of estimation. In order to investigate the effect of the identification error we first assume that the order of the model is known. The results are studied in section 8.6.1. Then we relax this assumption and examine the results for unknown model orders in section 8.6.2. Finally, the lead-time forecasts are compared in section 8.6.3. 8.6.1 INARMA with Known Order In this section, we first compare the one-step ahead forecasts produced by each M.Mohammadipour, 2009, Chapter 8 167 method (INARMA, Croston, SBA, and SBJ). The three-step and six-step ahead forecasts are then compared. The results show that INARMA almost always produces the lowest MSE for all four processes (INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1)) when all points in time are considered. This is expected because the INARMA one-step ahead forecasts are MMSE forecasts and therefore when the demand follows an INARMA process, INARMA forecasts should outperform the benchmarks in terms of MSE. The results also confirm that when only issue points are considered, the INARMA forecasts are biased (see Appendix 8.D). This is expected because the least squares criterion and therefore the CLS estimates are developed for the case where parameters are updated each period regardless of the demand being positive or zero. As parameters need to be estimated for INARMA models, with an increase in the number of observations, the forecasts? accuracy increases. The degree of improvement over benchmarks using the MSE measure, when all points in time are considered, is shown in Table 8-16 to Table ?8-21. It should be noted that in some tables ? is used in two different ways: when it is below the benchmark methods, it is the smoothing parameter for that specific method and when it is in the first column of the table, it is the autoregressive parameter of the INARMA process. Table 8-16 One-step ahead for INARMA(0,0) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.8691 0.9114 0.9150 0.9473 0.9711 0.9731 0.9740 0.9894 0.9904 0.9794 0.9872 0.9875 ?= 0.5 0.9502 0.9773 0.9789 0.9747 0.9895 0.9904 0.9762 0.9870 0.9876 0.9675 0.9769 0.9772 ?= 0.7 0.9694 0.9900 0.9917 0.9732 0.9857 0.9862 0.9706 0.9812 0.9814 0.9587 0.9688 0.9690 ?= 1 0.9749 0.9892 0.9894 0.9688 0.9789 0.9788 0.9575 0.9670 0.9668 0.9478 0.9572 0.9569 ?= 3 0.9596 0.9525 0.9484 0.9424 0.9369 0.9331 0.9323 0.9257 0.9218 0.9208 0.9136 0.9096 ?= 5 0.9498 0.9255 0.9178 0.9367 0.9112 0.9034 0.9269 0.9048 0.8974 0.9126 0.8889 0.8814 ?= 20 0.9529 0.8212 0.7940 0.9315 0.8023 0.7756 0.9281 0.7990 0.7721 0.9124 0.7851 0.7586 The results show that the improvement increases when more observations are available for higher values of ? (??0.7). With more observations, the accuracy of parameters? estimates and therefore forecasts of INARMA and benchmark methods BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 168 become more accurate. However, the results suggest that the accuracy of INARMA forecasts improves at a faster rate than the benchmarks. The simulation results also show that for INARMA(0,0) and INMA(1) processes, the improvement over benchmarks is not considerable. However, with the presence of an autoregressive component, as in the INAR(1) and INARMA(1,1) cases, the improvement is considerable for the cases in which the autoregressive parameter is high (?= 0.9). As can be seen from Table ?8-16, when ?= 20, the improvement of MSE of INARMA over SBA and SBJ is very high. This is because these methods are designed for highly intermittent demand, but when ?= 20, the demand is barely intermittent and the methods do not reduce to SES. In this case, Croston?s method is equivalent to SES. Table 8-17 One-step ahead for INMA(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9421 0.9733 0.9755 0.9634 0.9816 0.9829 0.9715 0.9833 0.9837 0.9583 0.9704 0.9709 ?= 0.5,?= 0.5 0.9124 0.9470 0.9490 0.9395 0.9645 0.9660 0.9400 0.9607 0.9620 0.9282 0.9483 0.9495 ?= 0.9,?= 0.5 0.9124 0.9543 0.9569 0.9227 0.9503 0.9521 0.9273 0.9503 0.9515 0.9118 0.9352 0.9367 ?= 0.1,?= 1 0.9739 0.9871 0.9871 0.9629 0.9752 0.9751 0.9566 0.9673 0.9671 0.9426 0.9532 0.9530 ?= 0.5,?= 1 0.9872 1.0027 1.0024 0.9685 0.9793 0.9785 0.9550 0.9674 0.9669 0.9425 0.9545 0.9539 ?= 0.9,?= 1 1.0070 1.0146 1.0129 0.9849 0.9945 0.9931 0.9750 0.9839 0.9825 0.9628 0.9706 0.9690 ?= 0.1,?= 3 0.9719 0.9611 0.9563 0.9587 0.9490 0.9442 0.9500 0.9390 0.9341 0.9334 0.9249 0.9204 ?= 0.5,?= 3 1.0174 0.9948 0.9870 1.0035 0.9805 0.9728 0.9954 0.9730 0.9655 0.9763 0.9536 0.9461 ?= 0.9,?= 3 1.0720 1.0282 1.0166 1.0657 1.0262 1.0150 1.0405 1.0045 0.9937 1.0300 0.9917 0.9809 ?= 0.1,?= 5 0.9788 0.9474 0.9383 0.9499 0.9252 0.9169 0.9445 0.9168 0.9082 0.9308 0.9028 0.8944 ?= 0.5,?= 5 1.0318 0.9787 0.9648 1.0241 0.9679 0.9537 1.0019 0.9461 0.9322 0.9866 0.9376 0.9244 ?= 0.9,?= 5 1.0969 1.0211 1.0025 1.0777 1.0051 0.9867 1.0635 0.9917 0.9736 1.0345 0.9622 0.9445 When data is produced by an INARMA(0,0) or an INMA(1) process, the results of INARMA forecasts are only compared to those of Croston, SBA and SBJ with smoothing parameter 0.2. But for INAR(1) and INARMA(1,1) processes where an autoregressive component is present, the benchmark methods with smoothing parameter 0.5 are also included in comparisons (Table ?8-19 and Table ?8-21). This is because when the autoregressive parameter is high, the benchmark methods with higher smoothing parameter produce better forecasts than those with smoothing parameter 0.2. BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 169 Table 8-18 One-step ahead with smoothing parameter 0.2 for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9752 1.0033 1.0026 0.9868 1.0061 1.0074 0.9747 0.9877 0.9884 0.9609 0.9726 0.9733 ?= 0.5,?= 0.5 0.7962 0.8190 0.8212 0.7715 0.7929 0.7941 0.7688 0.7852 0.7858 0.7408 0.7582 0.7590 ?= 0.9,?= 0.5 0.5848 0.5134 0.4988 0.5333 0.4666 0.4536 0.5251 0.4684 0.4559 0.4917 0.4396 0.4280 ?= 0.1,?= 1 0.9971 1.0153 1.0158 0.9746 0.9860 0.9858 0.9636 0.9725 0.9721 0.9433 0.9532 0.9529 ?= 0.5,?= 1 0.8845 0.8890 0.8870 0.8280 0.8308 0.8288 0.8122 0.8145 0.8124 0.7924 0.7962 0.7943 ?= 0.9,?= 1 0.5825 0.4598 0.4379 0.5478 0.4285 0.4078 0.5275 0.4196 0.3997 0.5068 0.4054 0.3865 ?= 0.1,?= 3 0.9922 0.9794 0.9742 0.9621 0.9539 0.9494 0.9538 0.9433 0.9385 0.9317 0.9222 0.9176 ?= 0.5,?= 3 0.9342 0.8928 0.8815 0.8807 0.8408 0.8301 0.8586 0.8180 0.8073 0.8356 0.7960 0.7857 ?= 0.9,?= 3 0.5985 0.3502 0.3175 0.5379 0.3104 0.2816 0.5215 0.3001 0.2723 0.4984 0.2848 0.2584 ?= 0.1,?= 5 0.9982 0.9644 0.9550 0.9675 0.9353 0.9261 0.9515 0.9224 0.9135 0.9325 0.9031 0.8943 ?= 0.5,?= 5 0.9170 0.8396 0.8217 0.8934 0.8194 0.8019 0.8586 0.7846 0.7677 0.8383 0.7676 0.7511 ?= 0.9,?= 5 0.5881 0.2457 0.2165 0.5374 0.2341 0.2067 0.5208 0.2322 0.2052 0.5022 0.2190 0.1934 Table 8-19 One-step ahead with smoothing parameter 0.5 for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.8996 0.9844 1.0063 0.8776 0.9584 0.9960 0.8577 0.9340 0.9680 0.8445 0.9192 0.9505 ?= 0.5,?= 0.5 0.7901 0.8582 0.7930 0.7459 0.8201 0.7663 0.7356 0.8014 0.7470 0.7034 0.7688 0.7252 ?= 0.9,?= 0.5 0.9191 0.3860 0.2257 0.8746 0.3646 0.2094 0.8416 0.3604 0.2104 0.8066 0.3418 0.1975 ?= 0.1,?= 1 0.8912 0.9544 0.9629 0.8619 0.9153 0.9243 0.8548 0.9036 0.9074 0.8318 0.8836 0.8915 ?= 0.5,?= 1 0.9069 0.8851 0.7594 0.8500 0.8284 0.7064 0.8316 0.8128 0.6949 0.8058 0.7894 0.6785 ?= 0.9,?= 1 0.9332 0.2532 0.1451 0.8807 0.2362 0.1349 0.8583 0.2296 0.1316 0.8248 0.2256 0.1294 ?= 0.1,?= 3 0.8630 0.8141 0.7700 0.8310 0.7906 0.7558 0.8293 0.7850 0.7434 0.8105 0.7676 0.7264 ?= 0.5,?= 3 0.9728 0.7309 0.5562 0.9190 0.6883 0.5230 0.8957 0.6672 0.5067 0.8703 0.6505 0.4946 ?= 0.9,?= 3 0.9337 0.1106 0.0630 0.8746 0.1003 0.0569 0.8514 0.0972 0.0552 0.8186 0.0922 0.0523 ?= 0.1,?= 5 0.8603 0.7393 0.6599 0.8312 0.7100 0.6363 0.8165 0.6998 0.6270 0.8010 0.6854 0.6142 ?= 0.5,?= 5 0.9588 0.5936 0.4247 0.9311 0.5748 0.4116 0.8952 0.5533 0.3949 0.8720 0.5369 0.3847 ?= 0.9,?= 5 0.9364 0.0642 0.0362 0.8728 0.0620 0.0350 0.8586 0.0612 0.0346 0.8171 0.0578 0.0326 The degree of improvement over benchmarks, using the MASE measure, is shown in Appendix 8.E. The results show that for INARMA(0,0) and INMA(1) processes, the improvement over benchmarks, in terms of MASE, is not considerable. But for INAR(1) and INARMA(1,1) processes, the improvement is high for the cases in which the autoregressive parameter is high. This confirms the results using MSE. BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 170 Table 8-20 One-step ahead with smoothing parameter 0.2 for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9881 1.0188 1.0202 0.9798 1.0007 1.0021 0.9716 0.9882 0.9892 0.9516 0.9667 0.9674 ?= 0.1,?= 0.9,?= 0.5 0.8263 0.8614 0.8633 0.8117 0.8382 0.8400 0.8272 0.8483 0.8495 0.8113 0.8311 0.8322 ?= 0.5,?= 0.5,?= 0.5 0.7641 0.7846 0.7846 0.7271 0.7439 0.7443 0.7000 0.7128 0.7130 0.6491 0.6638 0.6641 ?= 0.9,?= 0.1,?= 0.5 0.5621 0.4725 0.4577 0.5314 0.4674 0.4542 0.4982 0.4397 0.4276 0.4740 0.4208 0.4091 ?= 0.1,?= 0.1,?= 1 1.0022 1.0176 1.0176 0.9872 1.0013 1.0012 0.9657 0.9782 0.9780 0.9410 0.9530 0.9528 ?= 0.1,?= 0.9,?= 1 0.9076 0.9143 0.9127 0.8657 0.8765 0.8754 0.8613 0.8670 0.8655 0.8523 0.8595 0.8581 ?= 0.5,?= 0.5,?= 1 0.8216 0.8156 0.8119 0.7701 0.7676 0.7645 0.7533 0.7488 0.7456 0.7082 0.7073 0.7045 ?= 0.9,?= 0.1,?= 1 0.5736 0.4402 0.4180 0.5206 0.4057 0.3860 0.4975 0.3911 0.3723 0.4610 0.3665 0.3492 ?= 0.1,?= 0.1,?= 5 1.0117 0.9720 0.9613 0.9802 0.9453 0.9352 0.9653 0.9306 0.9206 0.9442 0.9096 0.8998 ?= 0.1,?= 0.9,?= 5 0.9149 0.8425 0.8257 0.8857 0.8149 0.7987 0.8560 0.7915 0.7762 0.8482 0.7853 0.7701 ?= 0.5,?= 0.5,?= 5 0.8271 0.7310 0.7106 0.7831 0.6908 0.6713 0.7541 0.6713 0.6531 0.7249 0.6439 0.6262 ?= 0.9,?= 0.1,?= 5 0.5341 0.2424 0.2140 0.5230 0.2214 0.1951 0.4948 0.2153 0.1901 0.8477 0.8073 0.0154 Table 8-21 One-step ahead with smoothing parameter 0.5 for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9038 1.0019 1.0255 0.8647 0.9572 0.9917 0.8475 0.9364 0.9709 0.8258 0.9106 0.9474 ?= 0.1,?= 0.9,?= 0.5 0.7641 0.8583 0.8595 0.7157 0.8114 0.8250 0.7357 0.8203 0.8208 0.7211 0.8043 0.8037 ?= 0.5,?= 0.5,?= 0.5 0.7960 0.8529 0.7388 0.7567 0.7983 0.6949 0.7268 0.7625 0.6618 0.6713 0.7108 0.6191 ?= 0.9,?= 0.1,?= 0.5 0.9343 0.3702 0.2060 0.8703 0.3592 0.2061 0.8319 0.3426 0.1959 0.7895 0.3213 0.1843 ?= 0.1,?= 0.1,?= 1 0.8982 0.9543 0.9513 0.8780 0.9369 0.9357 0.8621 0.9150 0.9112 0.8355 0.8885 0.8884 ?= 0.1,?= 0.9,?= 1 0.8859 0.8900 0.7964 0.8257 0.8452 0.7746 0.8359 0.8396 0.7565 0.8209 0.8294 0.7517 ?= 0.5,?= 0.5,?= 1 0.9256 0.8391 0.6537 0.8635 0.7895 0.6188 0.8433 0.7673 0.6014 0.7877 0.7255 0.5714 ?= 0.9,?= 0.1,?= 1 0.9236 0.2359 0.1346 0.8665 0.2240 0.1270 0.8322 0.2161 0.1228 0.7864 0.2054 0.1164 ?= 0.1,?= 0.1,?= 5 0.8940 0.7391 0.6392 0.8696 0.7208 0.6244 0.8625 0.7127 0.6135 0.8371 0.6939 0.6003 ?= 0.1,?= 0.9,?= 5 0.9156 0.5979 0.4408 0.8931 0.5789 0.4277 0.8616 0.5651 0.4186 0.8536 0.5581 0.4132 ?= 0.5,?= 0.5,?= 5 0.9496 0.4851 0.3221 0.8921 0.4558 0.3031 0.8605 0.4491 0.2998 0.8230 0.4266 0.2851 ?= 0.9,?= 0.1,?= 5 0.8881 0.0630 0.0357 0.8565 0.0576 0.0325 0.8252 0.0568 0.0320 0.9035 0.6340 0.5079 The results of comparing INARMA with benchmark methods for three-step ahead forecasts are given in Table ?8-22 to Table ?8-27 (See Appendix 8.F for the six-step ahead results). The results of comparing the h-step ahead forecasts (?= 3, 6) of INARMA with benchmarks for INARMA(0,0) are very close to the results of one- step ahead forecasts. BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 171 Table 8-22 Three-step ahead for INARMA(0,0) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.8459 0.8930 0.8980 0.9447 0.9692 0.9714 0.9751 0.9886 0.9898 0.9826 0.9901 0.9904 ?= 0.5 0.9458 0.9761 0.9785 0.9756 0.9910 0.9919 0.9735 0.9846 0.9851 0.9665 0.9757 0.9759 ?= 0.7 0.9693 0.9909 0.9924 0.9751 0.9868 0.9871 0.9720 0.9824 0.9827 0.9586 0.9686 0.9687 ?= 1 0.9758 0.9901 0.9903 0.9686 0.9791 0.9789 0.9644 0.9724 0.9720 0.9475 0.9564 0.9561 ?= 3 0.9578 0.9475 0.9432 0.9411 0.9346 0.9307 0.9305 0.9222 0.9182 0.9196 0.9137 0.9100 ?= 5 0.9522 0.9320 0.9249 0.9389 0.9149 0.9073 0.9263 0.9029 0.8954 0.9118 0.8908 0.8836 ?= 20 0.9549 0.8234 0.7960 0.9331 0.8011 0.7740 0.9265 0.7968 0.7699 0.9152 0.7911 0.7649 Table 8-23 Three-step ahead for INMA(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9359 0.9747 0.9782 0.9738 0.9916 0.9928 0.9660 0.9812 0.9820 0.9593 0.9711 0.9716 ?= 0.5,?= 0.5 0.9514 0.9890 0.9922 0.9541 0.9782 0.9798 0.9503 0.9713 0.9726 0.9280 0.9474 0.9486 ?= 0.9,?= 0.5 0.9481 0.9968 1.0009 0.9423 0.9752 0.9776 0.9297 0.9595 0.9615 0.9142 0.9367 0.9380 ?= 0.1,?= 1 0.9727 0.9880 0.9882 0.9664 0.9784 0.9783 0.9530 0.9666 0.9667 0.9357 0.9470 0.9468 ?= 0.5,?= 1 0.9614 0.9816 0.9820 0.9430 0.9580 0.9579 0.9274 0.9431 0.9431 0.9066 0.9214 0.9213 ?= 0.9,?= 1 0.9597 0.9746 0.9741 0.9269 0.9385 0.9377 0.9113 0.9312 0.9312 0.8884 0.9023 0.9017 ?= 0.1,?= 3 0.9542 0.9490 0.9449 0.9369 0.9290 0.9246 0.9246 0.9208 0.9169 0.9097 0.9042 0.9001 ?= 0.5,?= 3 0.9426 0.9304 0.9246 0.9146 0.9060 0.9007 0.8970 0.8871 0.8818 0.8797 0.8728 0.8677 ?= 0.9,?= 3 0.9444 0.9307 0.9238 0.9052 0.8900 0.8830 0.8915 0.8789 0.8723 0.8626 0.8496 0.8431 ?= 0.1,?= 5 0.9535 0.9341 0.9265 0.9339 0.9077 0.8996 0.9143 0.8929 0.8853 0.9010 0.8786 0.8710 ?= 0.5,?= 5 0.9401 0.9164 0.9067 0.9199 0.8872 0.8769 0.8997 0.8650 0.8546 0.8741 0.8418 0.8319 ?= 0.9,?= 5 0.9493 0.9035 0.8903 0.9075 0.8631 0.8503 0.8929 0.8528 0.8406 0.8618 0.8220 0.8101 Table 8-24 Three-step ahead with smoothing parameter 0.2 for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9779 1.0054 1.0076 0.9766 0.9924 0.9934 0.9688 0.9841 0.9850 0.9538 0.9672 0.9678 ?= 0.5,?= 0.5 1.0023 1.0387 1.0413 0.9757 1.0063 1.0083 0.9403 0.9708 0.9728 0.8924 0.9209 0.9228 ?= 0.9,?= 0.5 0.9722 0.9494 0.9365 0.9039 0.8583 0.8446 0.8577 0.8218 0.8098 0.7963 0.7682 0.7574 ?= 0.1,?= 1 0.9789 0.9993 1.0002 0.9623 0.9752 0.9752 0.9497 0.9607 0.9605 0.9347 0.9469 0.9469 ?= 0.5,?= 1 1.0005 1.0231 1.0231 0.9571 0.9759 0.9757 0.9256 0.9445 0.9444 0.8840 0.9012 0.9009 ?= 0.9,?= 1 0.9861 0.8725 0.8463 0.9031 0.8023 0.7786 0.8624 0.7758 0.7537 0.8057 0.7239 0.7029 ?= 0.1,?= 3 0.9586 0.9553 0.9512 0.9445 0.9363 0.9318 0.9253 0.9201 0.9161 0.9050 0.8995 0.8955 ?= 0.5,?= 3 0.9964 0.9755 0.9670 0.9330 0.9170 0.9095 0.9118 0.8958 0.8884 0.8730 0.8639 0.8573 ?= 0.9,?= 3 0.9680 0.6638 0.6176 0.8894 0.6050 0.5624 0.8873 0.6099 0.5665 0.8018 0.5559 0.5167 ?= 0.1,?= 5 0.9576 0.9332 0.9251 0.9296 0.9014 0.8931 0.9172 0.8933 0.8853 0.9035 0.8787 0.8707 ?= 0.5,?= 5 1.0170 0.9605 0.9454 0.9463 0.8926 0.8779 0.9114 0.8690 0.8558 0.8659 0.8225 0.8096 ?= 0.9,?= 5 0.9575 0.5535 0.5009 0.8854 0.8709 0.8602 0.8585 0.4990 0.4524 0.7998 0.4644 0.4210 BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 172 For an INMA(1) process, the performance of INARMA compared to benchmark methods is improved for h-step ahead forecasts compared to one-step ahead forecasts. This could be because the benchmark methods use the same forecast as one-step ahead forecast, but the INMA(1) method updates the forecasts. Table 8-25 Three-step ahead with smoothing parameter 0.5 for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.8885 0.9715 0.9774 0.8680 0.9432 0.9476 0.8537 0.9305 0.9356 0.8285 0.9070 0.9135 ?= 0.5,?= 0.5 0.9040 1.0122 1.0132 0.8470 0.9572 0.9595 0.8166 0.9260 0.9295 0.7684 0.8704 0.8741 ?= 0.9,?= 0.5 1.0808 0.7588 0.5810 1.0428 0.6862 0.5234 0.9866 0.6757 0.5192 0.9210 0.6331 0.4861 ?= 0.1,?= 1 0.8477 0.9174 0.9078 0.8337 0.8935 0.8821 0.8181 0.8761 0.8642 0.8076 0.8658 0.8543 ?= 0.5,?= 1 0.8654 0.9204 0.8876 0.8261 0.8750 0.8432 0.8023 0.8504 0.8206 0.7673 0.8105 0.7808 ?= 0.9,?= 1 1.1085 0.5490 0.3835 1.0376 0.5089 0.3552 1.0002 0.4991 0.3480 0.9346 0.4595 0.3190 ?= 0.1,?= 3 0.8006 0.7790 0.7181 0.7972 0.7646 0.7029 0.7724 0.7476 0.6896 0.7575 0.7331 0.6762 ?= 0.5,?= 3 0.8384 0.7401 0.6411 0.7946 0.7033 0.6100 0.7727 0.6852 0.5947 0.7383 0.6629 0.5761 ?= 0.9,?= 3 1.0951 0.2602 0.1617 1.0240 0.2353 0.1460 1.0154 0.2317 0.1432 0.9261 0.2125 0.1313 ?= 0.1,?= 5 0.7956 0.7032 0.6145 0.7698 0.6751 0.5888 0.7597 0.6681 0.5823 0.7511 0.6579 0.5729 ?= 0.5,?= 5 0.8714 0.6602 0.5333 0.8028 0.6011 0.4821 0.7715 0.5927 0.4782 0.7308 0.5574 0.4492 ?= 0.9,?= 5 1.1015 0.1672 0.0996 1.0322 0.7275 0.5586 1.0000 0.1529 0.0914 0.9226 0.1424 0.0852 Table 8-26 Three-step ahead with smoothing parameter 0.2 for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9621 0.9993 0.9992 0.9806 0.9994 1.0006 0.9730 0.9897 0.9907 0.9526 0.9673 0.9680 ?= 0.1,?= 0.9,?= 0.5 0.9515 0.9996 1.0031 0.9404 0.9728 0.9753 0.9278 0.9577 0.9597 0.8986 0.9229 0.9244 ?= 0.5,?= 0.5,?= 0.5 0.9945 1.0227 1.0239 0.9531 0.9855 0.9875 0.9317 0.9631 0.9651 0.8928 0.9228 0.9247 ?= 0.9,?= 0.1,?= 0.5 1.3727 1.2345 1.2092 1.5485 1.4763 1.4539 1.5901 1.5111 1.4876 1.7403 1.6794 1.6553 ?= 0.1,?= 0.1,?= 1 0.9927 1.0168 1.0178 0.9670 0.9810 0.9810 0.9561 0.9699 0.9698 0.9220 0.9359 0.9360 ?= 0.1,?= 0.9,?= 1 0.9532 0.9718 0.9715 0.9177 0.9347 0.9344 0.9064 0.9227 0.9224 0.8833 0.8989 0.8985 ?= 0.5,?= 0.5,?= 1 0.9933 1.0106 1.0097 0.9427 0.9644 0.9641 0.9201 0.9347 0.9336 0.8862 0.9034 0.9027 ?= 0.9,?= 0.1,?= 1 1.3867 1.2511 1.2147 1.5431 1.3756 1.3331 1.5887 1.4125 1.3700 1.7368 1.5550 1.5095 ?= 0.1,?= 0.1,?= 5 0.9832 0.9584 0.9495 0.9375 0.9126 0.9041 0.9181 0.8978 0.8898 0.8928 0.8704 0.8625 ?= 0.1,?= 0.9,?= 5 0.9710 0.9245 0.9105 0.9130 0.8713 0.8584 0.8894 0.8487 0.8365 0.8556 0.8185 0.8068 ?= 0.5,?= 0.5,?= 5 1.0173 0.9483 0.9303 0.9598 0.9053 0.8889 0.9245 0.8736 0.8579 0.8803 0.8285 0.8133 ?= 0.9,?= 0.1,?= 5 1.7356 0.9574 0.8627 1.6699 0.9654 0.8737 1.7683 1.0071 0.9128 1.7151 0.9770 0.8836 For INAR(1) and INARMA(1,1) processes, the performance of INARMA over the benchmark methods is improved compared to the one-step ahead case when the autoregressive parameter is low. But, as discussed in chapter 6, when the BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 173 autoregressive parameter is high, the fact that the forecasts converge to the mean of the process results in poor forecasts compared to the one-step ahead case. As explained in chapter 6, some authors suggest using different models for different horizons in order to improve forecast accuracy (Cox, 1961; Tiao and Xu, 1993; Kang, 2003). Table 8-27 Three-step ahead with smoothing parameter 0.5 for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.8645 0.9711 1.0200 0.8670 0.9500 0.9861 0.8402 0.9282 0.9735 0.8223 0.9071 0.9479 ?= 0.1,?= 0.9,?= 0.5 0.8371 0.9597 1.0175 0.7945 0.9047 0.9622 0.7757 0.8869 0.9431 0.7477 0.8483 0.9002 ?= 0.5,?= 0.5,?= 0.5 0.8855 0.9781 0.9853 0.8377 0.9372 0.9549 0.8185 0.9158 0.9341 0.7771 0.8723 0.8951 ?= 0.9,?= 0.1,?= 0.5 1.5506 0.9825 0.6679 1.8084 1.2198 0.8406 1.8603 1.2357 0.8578 2.0440 1.3820 0.9572 ?= 0.1,?= 0.1,?= 1 0.8588 0.9342 0.9719 0.8309 0.8943 0.9200 0.8203 0.8794 0.9068 0.7889 0.8504 0.8796 ?= 0.1,?= 0.9,?= 1 0.7944 0.8515 0.8782 0.7608 0.8154 0.8461 0.7513 0.8033 0.8338 0.7334 0.7820 0.8109 ?= 0.5,?= 0.5,?= 1 0.8670 0.9014 0.8896 0.8186 0.8594 0.8564 0.7895 0.8212 0.8123 0.7642 0.7988 0.7923 ?= 0.9,?= 0.1,?= 1 1.5546 0.7857 0.5145 1.7618 0.8418 0.5498 1.8280 0.8853 0.5789 2.0404 0.9911 0.6449 ?= 0.1,?= 0.1,?= 5 0.8070 0.7083 0.6504 0.7714 0.6736 0.6199 0.7510 0.6620 0.6146 0.7323 0.6432 0.5955 ?= 0.1,?= 0.9,?= 5 0.7803 0.6129 0.5297 0.7334 0.5807 0.5030 0.7137 0.5707 0.4969 0.6856 0.5474 0.4774 ?= 0.5,?= 0.5,?= 5 0.8725 0.6197 0.4987 0.8124 0.5889 0.4761 0.7889 0.5699 0.4608 0.7503 0.5401 0.4354 ?= 0.9,?= 0.1,?= 5 1.9454 0.2820 0.1644 1.9418 0.2908 0.1712 2.0688 0.3114 0.1832 1.9978 0.2951 0.1731 8.6.2 INARMA with Unknown Order It was assumed in section 8.6.1 that the order of the INARMA process is known. However, in reality this is not the case and the autoregressive and moving average orders of the model need to be identified. As discussed in chapter 4, when simulating a high number of replications, automated approaches such as AIC and BIC should be used for identification rather than sample autocorrelation (SACF) and sample partial autocorrelation functions (SPACF). In chapter 4, two identification procedures were suggested. A two-stage identification procedure is based on using the Ljung-Box test to distinguish between INARMA(0,0) and other processes and then using the AIC (or AICC for small sample sizes) to select among the other possible INARMA processes. On the other hand, the one-stage method only uses the AIC to select among all possible INARMA models including INARMA(0,0). BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 174 For simplicity, we first assume that data can be produced by either an INARMA(0,0) or an INAR(1) process. The results of identification among these two processes are summarized in section 8.6.2.1. As an alternative approach to identification, we suggest that the model with the highest order in the set of models (in this case INAR(1)) can be used for forecasting. The results are presented in section 8.6.2.2. Then we assume that data can be produced by any of the four processes. The results of identification based using two-stage and one-stage identification procedures are presented in section 8.6.2.3. The most-general-model approach is also tested and the results are analyzed in section 8.6.2.4. The results of treating all models as INAR(1) are compared to the benchmark methods in section 8.6.2.5. 8.6.2.1 Identification among Two Processes In this section, it is assumed that data is either INARMA(0,0) or INAR(1). As suggested by Jung and Tremayne (2003), in order to distinguish between the INARMA(0,0) (or an i.i.d. Poisson process) and INAR(1), we test if the data show a significant serial dependence or not. This is done using a portmanteau test called the Ljung-Box test explained in section 4.2.3. The test is based on a ??-statistic given by: ??=?(?+ 2)? ??????1?? 2 ? ?=1 Equation 8-15 where ? is the sample size, ? is the number of autocorrelation lags included in the statistic (we assume ?= 10), ?? is the sample autocorrelation at lag ?. The ? ? statistic can be used when a univariate model is fitted to a time series. It can be used as a lack-of-fit test for a departure from randomness. Under the null hypothesis that the model fit is adequate, the test statistic is asymptotically chi-square distributed. Results are presented for a significance level of 0.05. The results in terms of percentage of series for which the model is correctly identified are summarized in Table 8-28 and Table 8-29 for both Ljung-Box and AIC identification procedures. The results of Table 8-28 show identification with the M.Mohammadipour, 2009, Chapter 8 175 Ljung-Box test provides better results than with the AIC for INARMA(0,0) series. Comparing the results of Table 8-28 to the results by Jung and Tremayne (2003) shows that the ?? statistic produces similar results to those suggested in their study. The simulation results show that for those cases where an INARMA(0,0) is misidentified as an INAR(1) process, the estimated autoregressive parameter is close to zero. Table 8-28 The percentage of correct identification for INARMA(0,0) series Parameters Ljung-Box AIC ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?= 0.3 94.42 95.53 97.10 95.00 91.00 89.60 87.70 81.50 ?= 0.5 93.35 94.50 94.60 95.60 92.20 88.30 83.70 78.70 ?= 0.7 93.16 94.11 96.00 95.60 91.80 87.80 83.10 74.80 ?= 1 93.64 94.25 94.60 95.30 91.10 83.20 80.40 71.40 ?= 3 95.00 92.50 94.20 95.20 83.10 79.60 74.70 69.70 ?= 5 92.80 93.70 93.80 94.60 84.20 77.20 74.40 68.80 ?= 20 93.20 94.20 93.60 94.80 83.70 75.90 72.40 68.50 It can be seen from Table ?8-29 that, for an INAR(1) process, when the autoregressive parameter is small (?= 0.1), the model is often misidentified as INARMA(0,0). The AIC is always better than the Ljung-Box method. For high values of ?, the correct model in identified by the AIC in most cases. Both identification methods perform better when more observations are available. For high values of ? (?= 0.9) and ? (?= 96) the two identification methods are close. Table 8-29 The percentage of correct identification for INAR(1) series Parameters Ljung-Box AIC ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?= 0.1,?= 0.5 5.90 6.00 5.10 5.40 13.70 19.90 28.40 47.20 ?= 0.5,?= 0.5 15.40 23.90 34.30 71.60 49.00 73.60 86.00 98.60 ?= 0.9,?= 0.5 34.20 65.50 84.80 100.00 90.00 98.50 99.70 100.00 ?= 0.1,?= 1 6.50 7.20 6.90 6.30 14.90 26.10 35.40 52.20 ?= 0.5,?= 1 16.30 22.60 35.40 73.90 57.70 80.70 90.10 99.70 ?= 0.9,?= 1 35.10 63.30 86.50 99.80 89.80 98.30 100.00 100.00 ?= 0.1,?= 3 9.00 5.90 7.20 6.30 22.30 33.90 39.10 57.00 ?= 0.5,?= 3 14.50 23.90 34.40 71.30 62.70 83.40 92.50 99.60 ?= 0.9,?= 3 34.50 66.00 83.50 99.90 91.10 98.70 99.80 100.00 ?= 0.1,?= 5 7.40 6.50 4.80 7.70 22.80 37.10 40.10 59.40 ?= 0.5,?= 5 16.80 24.40 35.30 72.50 67.00 86.60 93.20 99.70 ?= 0.9,?= 5 33.90 64.40 83.60 99.70 91.10 98.50 99.70 100.00 M.Mohammadipour, 2009, Chapter 8 176 Because the INARMA(0,0) process is correctly identified in most of the cases, the forecast accuracy of the INARMA(0,0) with identification is close to that of the case when the order is known. The same is true for an INAR(1) process with high values of ? and ?. As a result, the performance of these two processes compared to benchmarks is similar to what was discussed in section 8.6.1. The accuracy results of INAR(1) forecasts using ME, MSE and MASE for the two identification methods are shown in Table ?8-30. Similar results for the case that the order of the model is known are provided in Table ?8-31 for comparison. The results of Table ?8-30 show that the AIC produces better forecasts than the Ljung- Box method in most of the cases, except for the case where ?= 0.1 and ? and ? are small. The results also show that when ?= 0.1, although the percentage of correct identification is small, INARMA with identification produces more accurate forecasts compared to the case where the order is known. This means that, in this case, using an INARMA(0,0) forecast based on the average of the previous observations produces better results than estimating ? and ? and forecasting using an INAR(1) model. However, when the autoregressive parameter is high (?= 0.9) and the number of observations is small (?= 24), although the percentage of correct identification is considerable, the difference between INARMA without identification and with identification for the Ljung-Box method is huge. This is expected because here a time series with high autocorrelation is wrongly identified as a series with no autocorrelation. Therefore, instead of putting a high weight on the last observation, the forecast is based on the INARMA(0,0) model which uses the simple average of all previous observations with equal weights. However, as the length of history increases, the percentage of correct identification also increases and the forecast accuracy of the two cases become very close. This is also true when comparing the Ljung-Box and AIC identification methods. For high values of ?, the latter is considerably better than the former due to the higher percentage of correct identification and the fact that misidentification of an INAR(1) process with high autoregressive parameter as an INARMA(0,0) has a huge penalty. M.Mohammadipour, 2009, Chapter 8 177 Table 8-30 Accuracy of INAR(1) forecasts for Ljung-Box and AIC identification procedures Parameters ?=?? ?=?? ?=?? ?=?? Ljung-Box AIC Ljung-Box AIC Ljung-Box AIC Ljung-Box AIC ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 -0.0105 0.6244 1.0977 0.0006 0.6343 1.1239 0.0064 0.6235 1.0925 0.0104 0.6038 1.0563 -0.0056 0.5928 1.0145 0.0084 0.6196 1.0273 -0.0024 0.5691 0.9758 -0.0038 0.5660 0.9690 ?= 0.5,?= 0.5 0.0047 1.0475 1.4131 0.0144 1.0372 1.3421 -0.0072 1.0201 1.2497 0.0102 0.9106 1.1900 0.0016 0.9858 1.2016 0.0023 0.8703 1.1342 0.0062 0.8633 1.1078 0.0039 0.7970 1.0655 ?= 0.9,?= 0.5 0.0547 2.7057 1.9921 0.0082 1.4192 1.4380 0.0188 2.0866 1.6381 -0.0122 1.1614 1.3010 -0.0193 1.5043 1.3871 -0.0223 1.0431 1.1720 -0.0058 1.0078 1.1428 -0.0091 1.0118 1.1512 ?= 0.1,?= 1 -0.0022 1.2865 0.9486 0.0063 1.3055 0.8974 -0.0019 1.1974 0.8462 -0.0011 1.2412 0.8653 -0.0053 1.1850 0.8292 -0.0045 1.1771 0.8355 0.0106 1.1682 0.8102 -0.0064 1.1372 0.7995 ?= 0.5,?= 1 0.0108 2.1835 1.1859 0.0187 1.9727 1.1119 0.0284 2.1543 1.1541 -0.0017 1.8151 1.0594 0.0131 1.9543 1.0778 0.0031 1.7068 1.0146 0.0040 1.6895 0.9885 0.0046 1.5869 0.9646 ?= 0.9,?= 1 0.0246 5.3777 1.7945 -0.0147 2.8294 1.3379 0.0015 4.2493 1.5135 -0.0247 2.3516 1.1871 -0.0190 2.8263 1.2406 -0.0047 2.1486 1.1096 -0.0084 2.0473 1.0789 -0.0035 2.0222 1.0807 ?= 0.1,?= 3 0.0226 3.9401 0.8599 0.0091 3.8786 0.8505 -0.0248 3.6785 0.8039 0.0103 3.8087 0.8416 0.0014 3.6089 0.8094 0.0014 3.6078 0.8150 0.0065 3.4860 0.7868 -0.0025 3.4883 0.7823 ?= 0.5,?= 3 0.0856 6.5598 1.1244 -0.0293 5.9107 1.0177 -0.0051 6.2115 1.0573 0.0050 5.4093 0.9960 0.0089 5.9927 1.0129 0.0133 5.1793 0.9548 0.0333 5.2292 0.9589 -0.0359 4.6773 0.9000 ?= 0.9,?= 3 0.1048 15.6396 1.6548 0.0026 8.1910 1.2077 -0.0602 11.6692 1.3805 0.0020 6.9766 1.1231 0.0335 9.4088 1.2402 -0.0010 6.5496 1.0791 0.0076 6.1685 1.0349 -0.0080 6.0951 1.0283 ?= 0.1,?= 5 -0.0021 6.6522 0.8473 0.0089 6.4835 0.8336 0.0023 6.1105 0.8153 0.0198 6.2402 0.8273 -0.0329 5.8728 0.7829 0.0073 6.0014 0.7918 -0.0209 5.7109 0.7718 0.0015 5.8188 0.7695 ?= 0.5,?= 5 0.0828 10.9849 1.1122 -0.0084 9.6383 1.0246 0.0380 10.2881 1.0358 -0.0419 8.7787 0.9678 0.0139 9.9559 1.0261 0.0012 8.4644 0.9364 -0.0196 8.4499 0.9316 -0.0009 7.9078 0.8969 ?= 0.9,?= 5 -0.0928 26.7365 1.6375 -0.0277 13.8802 1.2078 -0.0658 21.3689 1.4308 -0.0124 11.6831 1.1051 -0.0208 15.1822 1.2121 0.0341 10.8485 1.0563 0.0020 10.2432 1.0281 -0.0286 10.1068 1.0197 M.Mohammadipour, 2009, Chapter 8 178 Table 8-31 Accuracy of INAR(1) forecasts when the order is known Parameters ?=?? ?=?? ?=?? ?=?? ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 0.0026 0.6614 1.1418 -0.0089 0.6286 1.0555 0.0052 0.6047 1.0237 0.0045 0.5822 0.9822 ?= 0.5,?= 0.5 -0.0026 0.9466 1.3290 -0.0089 0.8743 1.1807 0.0044 0.8532 1.1570 -0.0054 0.7834 1.0534 ?= 0.9,?= 0.5 0.0017 1.2082 1.3901 0.0042 1.1489 1.2894 -0.0043 1.0811 1.2176 -0.0033 1.0164 1.1486 ?= 0.1,?= 1 -0.0149 1.2880 0.9376 -0.0059 1.2072 0.8496 0.0052 1.1967 0.8382 0.0029 1.1663 0.8057 ?= 0.5,?= 1 -0.0195 1.8846 1.0837 -0.0018 1.7244 1.0202 -0.0045 1.6745 1.0059 0.0026 1.5905 0.9618 ?= 0.9,?= 1 -0.0121 2.5216 1.2401 0.0054 2.2655 1.1594 -0.0146 2.1925 1.1262 -0.0105 2.0319 1.0686 ?= 0.1,?= 3 0.0100 3.9224 0.8544 -0.0176 3.6703 0.8228 -0.0010 3.6641 0.8136 0.0085 3.4701 0.7846 ?= 0.5,?= 3 -0.0308 5.6926 1.0126 -0.0035 5.1557 0.9491 -0.0078 4.9742 0.9331 0.0143 4.7749 0.9160 ?= 0.9,?= 3 -0.0906 7.5862 1.1509 -0.0298 6.7494 1.1026 -0.0243 6.4376 1.0658 -0.0093 6.0230 1.0205 ?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0209 6.1869 0.8143 0.0282 6.0095 0.7973 0.0139 5.7529 0.7742 ?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0082 8.6581 0.9653 0.0116 8.3583 0.9392 -0.0034 7.8699 0.9011 ?= 0.9,?= 5 -0.0013 11.9986 1.1483 -0.0126 11.2051 1.0914 -0.0467 10.8985 1.0720 0.0185 10.1102 1.0272 8.6.2.2 All-INAR(1) In this section, it is again assumed that data is produced by either an INARMA(0,0) or an INAR(1) process. Instead of identification, we assume that the most general model, INAR(1) in this case, is used for estimation and forecasting. It is expected that if the data is in fact an INARMA(0,0) process, the estimated autoregressive parameter should be close to zero and the results confirm this. In general, the forecasting accuracy deteriorates slightly in this case compared to the case of using Ljung-Box for identification among two possible models. The results for all points in time are shown in Table ?8-32. This shows that, instead of adding an extra step to the INARMA forecasting procedure for identification, treating everything as an INAR(1) process produces close results to those with identification and it has the advantage of being simple. The degree of deterioration caused by skipping identification is on average 2 percent for both MSE and MASE. M.Mohammadipour, 2009, Chapter 8 179 Table 8-32 Accuracy of forecasts with identification and all-INAR(1) for INARMA(0,0) series Parameters ?=?? ?=?? ?=?? ?=?? Ljung-Box All-INAR(1) Ljung-Box All-INAR(1) Ljung-Box All-INAR(1) Ljung-Box All-INAR(1) ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.3 0.0000 0.3509 1.1781 -0.0040 0.3610 1.1919 0.0005 0.3309 1.1476 -0.0008 0.3383 1.1552 -0.0032 0.3161 1.0892 -0.0066 0.3295 1.0741 -0.0039 0.3075 1.0115 0.0025 0.3131 1.0161 ?= 0.5 -0.0021 0.5796 1.0779 -0.0054 0.5967 1.0959 -0.0012 0.5512 1.0077 -0.0019 0.5613 1.0133 0.0009 0.5368 0.9725 0.0013 0.5561 0.9737 -0.0039 0.5183 0.9365 -0.0017 0.5191 0.9412 ?= 0.7 0.0003 0.8039 0.9807 -0.0062 0.8331 0.9786 -0.0039 0.7708 0.9005 -0.0017 0.7855 0.9152 0.0066 0.7572 0.8817 -0.0016 0.7510 0.8937 0.0048 0.7241 0.8537 0.0012 0.7374 0.8466 ?= 1 0.0012 1.1651 0.8716 -0.0028 1.1930 0.8947 0.0002 1.0979 0.8120 -0.0011 1.1178 0.8275 0.0000 1.0774 0.8025 -0.0068 1.0679 0.7956 0.0115 1.0490 0.7596 -0.0023 1.0481 0.7666 ?= 3 0.0210 3.4439 0.8065 -0.0273 3.5655 0.8128 -0.0154 3.3509 0.7703 0.0024 3.3711 0.7799 0.0087 3.2019 0.7589 -0.0083 3.2774 0.7651 0.0054 3.1061 0.7422 -0.0086 3.0890 0.7322 ?= 5 0.0099 5.8831 0.8153 -0.0158 5.8972 0.8071 -0.0067 5.5456 0.7624 -0.0150 5.4933 0.7567 0.0001 5.4289 0.7581 0.0010 5.4727 0.7642 0.0123 5.2099 0.7375 -0.0003 5.2125 0.7325 ?= 20 -0.0248 22.8486 0.7730 -0.0053 23.8074 0.8088 -0.0543 22.0121 0.7581 -0.0683 22.0701 0.7631 0.0183 21.2026 0.7420 -0.0101 22.2400 0.7589 0.0079 20.5646 0.7248 -0.0335 20.8631 0.7307 M.Mohammadipour, 2009, Chapter 8 180 8.6.2.3 Identification among Four Processes In this section it is assumed that data can be produced by one of the four processes that we focus on in this study: INARMA(0,0), INAR(1), INMA(1), or INARMA(1,1). As previously mentioned, two methods of identification are used. The two-stage method is based on first using the Ljung-Box ??-statistic of Equation 8-15 to distinguish between the INARMA(0,0) or random Poisson process from the other possible INARMA models. Then, the other three models (INAR(1), INMA(1), and INARMA(1,1)) are distinguished using the Akaike information criterion based on the expression for ARMA models: AIC??log?? 2 + 2? Equation 8-16 As explained in section 4.5, when the sample size is small (?/?< 40), the following bias correction is necessary: AICC ??log?? 2 + 2?+ 2?(?+ 1)/(????1) Equation 8-17 On the other hand, the one-stage method only uses the AIC to select the appropriate model. As discussed in chapter 4, although the above equations have been developed for ARMA models with a Normal distribution, as the likelihood function for INMA processes has not been established in the literature and AIC is a method of identification that can be automated, we use these equations to test the performance of AIC for INARMA processes. The percentage of correct identification for each of the four INARMA processes for both two-stage and one-stage methods is shown in Table ?8-33, Table ?8-34, Table ?8-35, and Table ?8-36. The results of Table ?8-33 confirm the results of section 8.6.2.1 in that the two-stage method identifies the INARMA(0,0) processes more frequently than the one-stage method. M.Mohammadipour, 2009, Chapter 8 181 Table 8-33 The percentage of correct identification for INARMA(0,0) series Parameters Two-stage identification One-stage identification ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?= 0.3 94.62 95.87 96.30 94.90 89.40 89.00 88.60 80.80 ?= 0.5 93.29 94.45 94.40 95.10 91.90 88.10 84.30 73.30 ?= 0.7 92.81 94.07 95.30 95.90 88.80 82.80 78.80 69.00 ?= 1 93.54 94.32 94.10 95.10 90.40 79.60 76.00 67.20 ?= 3 93.40 92.40 94.40 94.70 82.70 72.50 65.40 57.00 ?= 5 92.90 93.40 92.80 93.50 78.50 70.90 61.70 55.00 ?= 20 92.70 93.40 94.10 95.30 76.20 66.00 61.00 51.40 For an INAR(1) case, the results of Table ?8-34 confirm that when the autoregressive parameter is low, the process is misidentified in most cases for both identification methods. However, the one-stage method produces better results. On the other hand, with a high autoregressive parameter, the performance of both methods improves. The results also show that when more observations are available, the percentage of correct identification increases for both methods. The one-stage method outperforms the two-stage method in most of the cases, especially for small samples. Table 8-34 The percentage of correct identification for INAR(1) series Parameters Two-stage identification One-stage identification ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?= 0.1,?= 0.5 2.30 2.50 2.60 2.90 9.60 15.80 21.20 26.40 ?= 0.5,?= 0.5 9.50 19.80 25.80 52.70 38.40 55.60 67.70 75.80 ?= 0.9,?= 0.5 30.60 55.00 61.50 73.80 73.30 71.70 71.70 72.50 ?= 0.1,?= 1 1.70 1.50 1.80 3.00 13.20 19.20 22.20 30.00 ?= 0.5,?= 1 10.90 18.70 26.60 49.70 46.90 59.20 65.90 72.40 ?= 0.9,?= 1 28.60 52.40 62.50 71.50 74.70 71.80 72.20 69.30 ?= 0.1,?= 3 1.90 2.50 1.70 2.90 16.30 22.10 22.40 28.00 ?= 0.5,?= 3 11.20 16.10 24.30 50.20 47.40 57.90 60.70 68.00 ?= 0.9,?= 3 24.40 46.30 63.90 75.30 75.50 74.90 72.30 75.50 ?= 0.1,?= 5 2.70 2.40 2.70 2.50 17.50 22.70 22.50 26.10 ?= 0.5,?= 5 12.10 17.60 26.30 44.70 50.20 59.10 61.50 68.30 ?= 0.9,?= 5 27.30 49.40 63.00 73.80 76.20 75.40 73.30 73.90 As can be seen from Table ?8-35, an INMA(1) process is misidentified in most of the cases regardless of the size of the moving average parameter. However, the results show that it does not affect the forecasting accuracy to a great extent. This can be seen by comparing the results of Table ?8-38 and Table ?8-41. The one-stage identification method again outperforms the two-stage method. M.Mohammadipour, 2009, Chapter 8 182 Table 8-35 The percentage of correct identification for INMA(1) series Parameters Two-stage identification One-stage identification ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?= 0.1,?= 0.5 3.30 2.70 3.30 3.10 4.10 6.70 8.10 12.00 ?= 0.5,?= 0.5 4.30 4.00 4.60 7.60 13.40 19.80 23.00 23.20 ?= 0.9,?= 0.5 7.90 10.60 11.60 32.50 21.70 33.70 37.80 45.10 ?= 0.1,?= 1 4.40 3.90 3.30 1.50 5.60 6.80 8.10 11.50 ?= 0.5,?= 1 3.20 3.90 3.30 4.10 9.80 14.60 17.20 16.80 ?= 0.9,?= 1 7.80 7.60 6.70 14.90 19.80 23.50 24.70 21.50 ?= 0.1,?= 3 3.80 3.40 2.50 1.20 5.40 6.70 9.00 14.00 ?= 0.5,?= 3 2.00 2.70 2.50 2.10 10.90 14.60 15.10 11.50 ?= 0.9,?= 3 4.60 4.00 4.30 7.10 18.30 16.70 14.40 11.00 ?= 0.1,?= 5 4.80 2.30 2.50 2.00 5.70 7.40 12.10 12.10 ?= 0.5,?= 5 3.90 3.20 2.20 1.80 10.60 14.20 11.20 9.60 ?= 0.9,?= 5 3.80 4.30 3.20 5.00 17.30 15.80 13.40 7.50 The results of Table ?8-36 suggest that, as expected, when the autoregressive parameter is high, the correct model is identified more often than the case with low autoregressive parameter. The identification performance improves when the length of history increases. The one-stage identification method produces better results than the two-stage method in most of the cases. Table 8-36 The percentage of correct identification for INARMA(1,1) series Parameters Two-stage identification One-stage identification ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?= 0.1,?= 0.1,?= 0.5 0.60 0.40 0.90 1.70 1.30 2.40 4.40 13.20 ?= 0.1,?= 0.9,?= 0.5 1.70 2.70 6.90 14.00 5.80 11.20 10.80 12.00 ?= 0.5,?= 0.5,?= 0.5 3.10 6.20 10.80 14.50 7.80 12.10 13.90 16.90 ?= 0.9,?= 0.1,?= 0.5 6.20 15.50 19.60 29.50 14.20 23.70 26.10 25.60 ?= 0.1,?= 0.1,?= 1 0.10 0.80 1.30 1.70 1.70 6.90 11.80 20.10 ?= 0.1,?= 0.9,?= 1 2.30 4.20 7.10 11.20 7.90 14.40 15.30 12.80 ?= 0.5,?= 0.5,?= 1 3.00 8.60 11.80 15.20 11.50 18.80 20.10 14.70 ?= 0.9,?= 0.1,?= 1 7.30 20.00 29.70 31.30 24.20 31.60 34.70 34.40 ?= 0.1,?= 0.1,?= 5 0.50 1.70 2.20 3.70 8.80 18.70 23.70 33.10 ?= 0.1,?= 0.9,?= 5 7.00 13.20 17.20 23.80 29.50 40.40 42.30 27.20 ?= 0.5,?= 0.5,?= 5 12.20 18.80 32.50 37.90 35.60 46.30 50.40 43.40 ?= 0.9,?= 0.1,?= 5 14.20 31.40 42.10 47.80 37.10 49.50 48.70 47.80 The accuracy of INAR(1), INMA(1) and INARMA(1,1) forecasts using ME, MSE and MASE for two identification methods are presented in Table ?8-37, Table ?8-38, and Table ?8-39. Similar results for the cases that the order of the model is known are M.Mohammadipour, 2009, Chapter 8 183 provided in Table ?8-40, Table ?8-41, and Table ?8-42 for comparison reasons. The results show that for processes with an AR component, when the autoregressive parameter is high, misidentification has a great effect on the accuracy of forecasts. However, because the one-stage identification method identifies the correct model more frequently than the two-stage method, the forecasts are closer to the case of known order. But when the autoregressive parameter is small, the effect of misidentification on forecasting accuracy is also small. For INARMA processes without an AR component, the effect of misidentification on forecasting accuracy is small, regardless of the size of the MA parameter. When the number of observations increases, forecasts with identification and without identification have similar accuracy. M.Mohammadipour, 2009, Chapter 8 184 Table 8-37 Accuracy of INAR(1) forecasts for one-stage and two-stage identification procedures Parameters ?=?? ?=?? ?=?? ?=?? Two-stage identification One-stage identification Two-stage identification One-stage identification Two-stage identification One-stage identification Two-stage identification One-stage identification ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 -0.0013 0.6357 1.1459 -0.0005 0.6347 1.1208 0.0073 0.6327 1.1044 0.0115 0.6260 1.1063 0.0008 0.6026 1.0373 0.0056 0.6005 1.0295 0.0019 0.5765 0.9804 0.0010 0.5826 0.9799 ?= 0.5,?= 0.5 0.0034 1.1192 1.4704 0.0381 1.0614 1.3967 0.0278 1.0414 1.2999 0.0050 0.9311 1.2226 0.0179 1.0165 1.2303 0.0064 0.8655 1.1425 0.0200 0.8786 1.1276 -0.0055 0.8014 1.0694 ?= 0.9,?= 0.5 0.0283 2.6093 1.9312 0.0279 1.4243 1.4776 -0.0005 2.0101 1.6350 0.0034 1.1610 1.2703 0.0090 1.6472 1.4275 -0.0103 1.0725 1.2157 0.0025 1.0257 1.1445 -0.0120 0.9981 1.1395 ?= 0.1,?= 1 0.0188 1.3041 0.9162 0.0063 1.3144 0.9415 -0.008 1.2309 0.8614 -0.0024 1.2606 0.8758 0.0029 1.2061 0.8395 0.0071 1.2258 0.8523 0.0005 1.1699 0.8088 0.0044 1.1590 0.8104 ?= 0.5,?= 1 0.0409 2.3062 1.2365 0.0196 2.0526 1.1509 0.0131 2.0748 1.1167 0.0117 1.8415 1.0550 0.0019 1.9812 1.0876 0.0093 1.7273 1.0262 0.0109 1.7521 1.0034 0.0039 1.6017 0.9651 ?= 0.9,?= 1 -0.018 5.2825 1.7681 -0.0212 2.7024 1.2864 0.0258 4.2144 1.4907 0.0037 2.2686 1.1378 0.0222 3.1025 1.2932 -0.0043 2.1714 1.1203 0.0087 2.0297 1.0845 -0.0012 2.0015 1.0693 ?= 0.1,?= 3 -0.0055 3.7732 0.8183 -0.0310 3.8137 0.8345 -0.0096 3.7235 0.8334 0.0042 3.7900 0.8337 -0.0036 3.5162 0.7995 -0.0041 3.6297 0.8083 0.0098 3.5198 0.7902 -0.0184 3.4845 0.7845 ?= 0.5,?= 3 -0.0356 6.6692 1.0898 0.0256 6.0453 1.0472 0.0322 6.4381 1.0708 0.0170 5.5157 0.9955 0.0377 6.0613 1.0365 -0.0103 5.1944 0.9556 0.0263 5.1238 0.9517 0.0027 4.7714 0.9155 ?= 0.9,?= 3 0.0466 17.2237 1.6973 -0.0198 8.0490 1.2229 0.0386 12.4798 1.4291 0.0090 6.7598 1.1070 0.0380 8.8346 1.1905 0.0301 6.4818 1.0660 0.0107 6.1671 1.0259 0.0099 6.0402 1.0160 ?= 0.1,?= 5 -0.0215 6.3926 0.8466 0.0440 6.7467 0.8632 0.0314 6.1703 0.8199 0.0022 6.1377 0.8190 -0.0108 6.0691 0.8034 -0.0241 6.1470 0.8087 0.0073 5.7416 0.7720 -0.0252 5.9063 0.7726 ?= 0.5,?= 5 -0.0479 10.7311 1.0682 -0.0116 9.9783 1.0548 0.1183 10.4885 1.0480 -0.0001 9.0103 0.9843 -0.0082 9.7144 1.0139 -0.0479 8.4696 0.9420 0.0108 8.5805 0.9368 -0.0078 7.9000 0.9092 ?= 0.9,?= 5 -0.0995 26.0358 1.6186 0.1313 13.5067 1.1973 0.0334 21.0114 1.4054 0.0554 11.3171 1.0809 -0.0524 15.4657 1.1925 0.0363 10.8754 1.0706 -0.0140 10.2025 1.0191 -0.0036 10.0731 1.0171 M.Mohammadipour, 2009, Chapter 8 185 Table 8-38 Accuracy of INMA(1) forecasts for one-stage and two-stage identification procedures Parameters ?=?? ?=?? ?=?? ?=?? Two-stage identification One-stage identification Two-stage identification One-stage identification Two-stage identification One-stage identification Two-stage identification One-stage identification ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 0.0263 0.6370 1.1790 0.0006 0.6474 1.1537 0.0081 0.6080 1.0762 -0.0042 0.6022 1.0458 0.0030 0.5913 1.0025 0.0005 0.5732 0.9729 0.0064 0.5719 0.9804 -0.0088 0.6604 0.7515 ?= 0.5,?= 0.5 0.0166 0.8479 1.3582 0.0210 0.8392 1.2970 0.0125 0.8181 1.2071 -0.0001 0.7935 1.1635 0.0022 0.7871 1.1224 0.0013 0.7284 1.0496 0.0082 0.7597 1.0724 0.0401 0.9122 0.9183 ?= 0.9,?= 0.5 0.0246 1.0729 1.4336 -0.0101 1.9689 1.1383 0.0207 1.0169 1.3105 0.0269 0.9813 1.2952 0.0240 1.0123 1.2547 0.0304 0.8882 1.1203 0.0250 0.9228 1.1422 0.1589 1.0747 1.0178 ?= 0.1,?= 1 -0.0003 1.2975 0.9130 -0.0104 1.2574 0.8783 -0.0070 1.2071 0.8535 -0.0014 1.2389 0.8692 0.0017 1.1671 0.8266 -0.0103 1.1444 0.8014 0.0036 1.1466 0.8073 -0.0335 1.2159 0.8080 ?= 0.5,?= 1 0.0098 1.7287 1.0880 0.0241 1.7225 1.0816 0.0153 1.6183 1.0208 -0.0065 1.5827 0.9817 -0.0051 1.6029 1.0149 -0.0020 1.4386 0.9239 0.0077 1.5257 0.9653 0.0177 1.5554 0.9547 ?= 0.9,?= 1 0.0081 2.1437 1.2136 -0.0131 1.9568 1.1574 0.0027 2.0207 1.1294 0.0036 1.8299 1.0827 0.0221 2.0155 1.1022 0.0179 1.6855 0.9954 0.0025 1.7263 1.0134 0.0595 1.7732 1.0167 ?= 0.1,?= 3 -0.0143 3.8219 0.8451 0.0170 3.9592 0.8653 0.0095 3.6174 0.8141 -0.0217 3.7610 0.8249 -0.0132 3.5934 0.8019 0.0052 3.4847 0.7859 -0.0158 3.3791 0.7756 0.0006 3.5011 0.8056 ?= 0.5,?= 3 0.0029 5.1085 0.9854 0.0143 5.2555 0.9867 0.0086 4.8587 0.9522 -0.0237 4.8094 0.9361 0.0259 4.7425 0.9197 -0.0040 4.2445 0.8681 0.0024 4.5513 0.8899 -0.0040 4.2567 0.8749 ?= 0.9,?= 3 -0.0104 6.3963 1.1064 -0.0283 6.0747 1.0769 0.0162 5.9595 1.0526 0.0190 5.4653 1.0106 0.0191 5.7408 1.0273 -0.0034 4.8673 0.9248 -0.0221 5.0070 0.9420 -0.0028 4.8710 0.9265 ?= 0.1,?= 5 0.0349 6.4620 0.8532 -0.0388 6.5789 0.8436 -0.0228 6.1804 0.8130 -0.0241 6.1703 0.8007 -0.0035 5.9428 0.7934 0.0168 5.7806 0.7811 0.0003 5.7474 0.7794 0.0160 5.7818 0.7843 ?= 0.5,?= 5 0.0000 8.3916 0.9442 -0.0841 8.5133 0.9644 0.0077 8.0429 0.9225 -0.0184 7.8372 0.9231 0.0038 8.0922 0.9232 0.0028 7.1503 0.8623 0.0291 7.5896 0.8869 0.0027 7.1503 0.8628 ?= 0.9,?= 5 0.0320 10.5829 1.1034 -0.0552 9.9051 1.0576 -0.0166 10.2863 1.0590 0.0005 8.9502 0.9893 -0.0596 9.3927 0.9941 0.0135 7.9759 0.9193 -0.0112 8.3874 0.9427 0.0135 7.9764 0.9194 M.Mohammadipour, 2009, Chapter 8 186 Table 8-39 Accuracy of INARMA(1,1) forecasts for one-stage and two-stage identification procedures Parameters ?=?? ?=?? ?=?? ?=?? Two-stage identification One-stage identification Two-stage identification One-stage identification Two-stage identification One-stage identification Two-stage identification One-stage identification ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.1,?= 0.5 0.0011 0.7193 1.1489 0.0064 0.7234 1.2251 0.0071 0.6856 1.0945 0.0163 0.6807 1.1061 0.0040 0.6758 1.0704 0.0071 0.6736 1.0554 0.0059 0.6447 1.0204 -0.0001 0.6425 1.0050 ?= 0.1,?= 0.9,?= 0.5 0.0368 1.3416 1.5159 0.0243 1.2303 1.4021 0.0201 1.2114 1.2766 0.0094 1.0992 1.2406 0.0173 1.1478 1.2195 0.0162 1.0410 1.1899 0.0159 1.0410 1.1174 0.0138 1.0175 1.1117 ?= 0.5,?= 0.5,?= 0.5 0.0536 1.9040 1.5660 0.0198 1.6235 1.4007 0.0186 1.6852 1.3458 0.0318 1.4292 1.2942 0.0301 1.5869 1.3092 0.0198 1.3254 1.1971 0.0064 1.1989 1.0976 0.0049 1.1844 1.1020 ?= 0.9,?= 0.1,?= 0.5 0.1342 2.8645 2.0436 0.0480 1.5286 1.4458 0.0377 2.1973 1.6213 0.0289 1.3400 1.3159 0.0277 1.6027 1.3899 0.0310 1.1833 1.2270 0.0105 1.1303 1.1566 0.0057 1.1247 1.1435 ?= 0.1,?= 0.1,?= 1 0.0004 1.4375 0.9427 -0.0167 1.4091 0.9407 0.0130 1.3668 0.8973 0.0079 1.3705 0.9013 -0.0038 1.3390 0.8764 0.0020 1.3314 0.8871 -0.0064 1.2803 0.8575 -0.0080 1.2693 0.8538 ?= 0.1,?= 0.9,?= 1 0.0137 2.5594 1.2132 0.0161 2.3860 1.1767 -0.0010 2.4152 1.1409 -0.0137 2.1568 1.0963 0.0109 2.3587 1.0905 0.0001 1.9950 1.0265 0.0028 2.0246 1.0093 0.0032 1.9137 0.9745 ?= 0.5,?= 0.5,?= 1 0.0028 3.7251 1.3345 -0.0434 3.0511 1.1857 -0.0317 3.4130 1.2229 0.0026 2.7933 1.1045 0.0225 3.0826 1.1397 0.0027 2.5683 1.0629 -0.0053 2.3995 1.0005 -0.0111 2.2977 0.9876 ?= 0.9,?= 0.1,?= 1 0.0911 6.0831 1.7798 -0.0259 3.0317 1.2791 0.0230 4.8164 1.5115 -0.0462 2.4860 1.1595 -0.0248 3.4115 1.2711 -0.0062 2.3640 1.0960 -0.0036 2.2186 1.0714 -0.0052 2.1915 1.0725 ?= 0.1,?= 0.1,?= 5 -0.0126 7.1217 0.8862 0.0116 7.3120 0.8927 0.0082 6.8366 0.8465 -0.0300 6.7345 0.8344 0.0131 6.6983 0.8327 0.0088 6.6701 0.8331 -0.0071 6.4563 0.8119 -0.0136 6.3122 0.8035 ?= 0.1,?= 0.9,?= 5 0.0523 12.8778 1.1181 0.0211 11.8098 1.0700 0.0125 11.8990 1.0497 -0.0111 10.4597 0.9831 0.0224 11.3640 1.0295 -0.0163 9.9553 0.9482 -0.0102 9.9313 0.9337 -0.0177 9.2879 0.9064 ?= 0.5,?= 0.5,?= 5 -0.0698 18.7993 1.2511 -0.0942 14.7010 1.0942 0.0161 16.7775 1.1609 -0.0720 12.9606 1.0033 -0.0238 15.2377 1.0874 -0.0601 12.3897 0.9876 0.0213 11.8909 0.9624 -0.0353 11.4452 0.9463 ?= 0.9,?= 0.1,?= 5 0.0827 29.4367 1.6336 0.0109 15.2016 1.2041 -0.0115 23.6024 1.4174 -0.0590 12.3696 1.0847 -0.0741 16.4158 1.2023 -0.0188 11.9975 1.0709 -0.0215 11.0569 1.0183 -0.0287 11.1985 1.0193 M.Mohammadipour, 2009, Chapter 8 187 Table 8-40 Accuracy of INAR(1) forecasts when the order in known Parameters ?=?? ?=?? ?=?? ?=?? ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 0.0026 0.6614 1.1418 -0.0089 0.6286 1.0555 0.0052 0.6047 1.0237 0.0045 0.5822 0.9822 ?= 0.5,?= 0.5 -0.0026 0.9466 1.3290 -0.0089 0.8743 1.1807 0.0044 0.8532 1.1570 -0.0054 0.7834 1.0534 ?= 0.9,?= 0.5 0.0017 1.2082 1.3901 0.0042 1.1489 1.2894 -0.0043 1.0811 1.2176 -0.0033 1.0164 1.1486 ?= 0.1,?= 1 -0.0149 1.2880 0.9376 -0.0059 1.2072 0.8496 0.0052 1.1967 0.8382 0.0029 1.1663 0.8057 ?= 0.5,?= 1 -0.0195 1.8846 1.0837 -0.0018 1.7244 1.0202 -0.0045 1.6745 1.0059 0.0026 1.5905 0.9618 ?= 0.9,?= 1 -0.0121 2.5216 1.2401 0.0054 2.2655 1.1594 -0.0146 2.1925 1.1262 -0.0105 2.0319 1.0686 ?= 0.1,?= 3 0.0100 3.9224 0.8544 -0.0176 3.6703 0.8228 -0.0010 3.6641 0.8136 0.0085 3.4701 0.7846 ?= 0.5,?= 3 -0.0308 5.6926 1.0126 -0.0035 5.1557 0.9491 -0.0078 4.9742 0.9331 0.0143 4.7749 0.9160 ?= 0.9,?= 3 -0.0906 7.5862 1.1509 -0.0298 6.7494 1.1026 -0.0243 6.4376 1.0658 -0.0093 6.0230 1.0205 ?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0209 6.1869 0.8143 0.0282 6.0095 0.7973 0.0139 5.7529 0.7742 ?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0082 8.6581 0.9653 0.0116 8.3583 0.9392 -0.0034 7.8699 0.9011 ?= 0.9,?= 5 -0.0013 11.9986 1.1483 -0.0126 11.2051 1.0914 -0.0467 10.8985 1.0720 0.0185 10.1102 1.0272 Table 8-41 Accuracy of INMA(1) forecasts when the order in known Parameters ?=?? ?=?? ?=?? ?=?? ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 -0.0031 0.6295 1.1297 0.0027 0.6035 1.0473 0.0114 0.5993 1.0493 0.0007 0.5702 0.9736 ?= 0.5,?= 0.5 0.0395 0.8793 1.3407 0.0279 0.8552 1.2527 0.0229 0.7997 1.1494 0.0031 0.7885 1.0948 ?= 0.9,?= 0.5 0.0767 1.1019 1.5472 0.0671 1.0609 1.3632 0.0628 1.0229 1.2606 0.0445 0.9878 1.1797 ?= 0.1,?= 1 0.0234 1.2748 0.9163 0.0000 1.2038 0.8547 0.0082 1.1724 0.8365 0.0048 1.1313 0.7957 ?= 0.5,?= 1 0.0347 1.7554 1.0666 0.0440 1.6455 1.0247 0.0310 1.6074 1.0005 0.0204 1.5302 0.9648 ?= 0.9,?= 1 0.1249 2.2869 1.2485 0.1025 2.1650 1.1687 0.1148 2.0762 1.1223 0.1116 1.9944 1.0920 ?= 0.1,?= 3 0.0452 3.9039 0.8547 0.0402 3.6622 0.8158 0.0150 3.5353 0.7960 0.0062 3.4237 0.7767 ?= 0.5,?= 3 0.1260 5.2415 0.9971 0.0781 4.9565 0.9450 0.0929 4.8942 0.9424 0.0608 4.6821 0.9036 ?= 0.9,?= 3 0.2970 6.7422 1.1268 0.3209 6.5993 1.1023 0.2796 6.2599 1.0587 0.2718 6.0678 1.0364 ?= 0.1,?= 5 0.0838 6.4549 0.8431 0.0132 6.2194 0.8182 0.0226 5.9292 0.7937 0.0111 5.7446 0.7806 ?= 0.5,?= 5 0.1407 8.7278 0.9676 0.1984 8.3020 0.9417 0.1738 8.0993 0.9245 0.1269 7.8404 0.9063 ?= 0.9,?= 5 0.4387 11.6802 1.1864 0.4081 10.9042 1.0713 0.4804 10.7677 1.0895 0.4479 10.1863 1.0242 Table 8-42 Accuracy of INARMA(1,1) forecasts when the order in known Parameters ?=?? ?=?? ?=?? ?=?? ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.1,?= 0.5 -0.0007 0.7482 1.2163 -0.0066 0.7138 1.1186 0.0019 0.6777 1.0787 -0.0047 0.6468 1.0076 ?= 0.1,?= 0.9,?= 0.5 0.0401 1.2112 1.4173 0.0336 1.1340 1.2974 0.0267 1.0734 1.1958 0.0163 1.0429 1.1248 ?= 0.5,?= 0.5,?= 0.5 0.0485 1.5791 1.4864 0.0209 1.4070 1.2666 0.0445 1.3825 1.1924 0.0261 1.2261 1.1137 ?= 0.9,?= 0.1,?= 0.5 0.0920 1.4003 1.4228 0.0532 1.2843 1.3142 0.0487 1.2226 1.2608 0.0324 1.1110 1.1489 ?= 0.1,?= 0.1,?= 1 -0.0041 1.4908 0.9689 -0.0108 1.4146 0.9145 -0.0129 1.3358 0.8925 -0.0119 1.2833 0.8575 ?= 0.1,?= 0.9,?= 1 0.0192 2.3265 1.1462 -0.0185 2.2569 1.0930 0.0368 2.2224 1.0713 0.0149 2.0900 1.0210 ?= 0.5,?= 0.5,?= 1 0.0296 3.1322 1.1914 0.0250 2.7101 1.1033 0.0409 2.6748 1.0798 0.0220 2.4363 1.0245 ?= 0.9,?= 0.1,?= 1 0.0665 2.8324 1.2617 0.0494 2.5623 1.1748 0.0640 2.4301 1.1219 0.0413 2.2588 1.0786 ?= 0.1,?= 0.1,?= 5 0.0279 7.4165 0.8824 -0.0070 6.9501 0.8502 0.0011 6.7534 0.8307 0.0006 6.4253 0.8079 ?= 0.1,?= 0.9,?= 5 0.0436 11.2147 1.0506 0.0580 10.5085 0.9941 0.0521 10.1101 0.9589 0.0606 9.7366 0.9338 ?= 0.5,?= 0.5,?= 5 0.1082 14.0754 1.0425 0.1239 13.0227 1.0240 0.1074 12.7060 0.9952 0.1065 11.8259 0.9556 ?= 0.9,?= 0.1,?= 5 0.1214 13.8720 1.1795 0.1895 12.4636 1.0932 0.1569 12.0389 1.0813 0.0906 11.2782 1.0311 M.Mohammadipour, 2009, Chapter 8 188 8.6.2.4 All-INARMA(1,1) In this section the method of section 8.6.2.2 is extended to include all four processes. Therefore data can be produced by either an INARMA(0,0), INAR(1), INMA(1) or an INARMA(1,1) process. Then, for estimation of parameters and forecasting, an INARMA(1,1) process is used. We expect that when data is in fact INARMA(0,0) the estimated autoregressive and moving average parameters (?,?) will be close to zero, and for INAR(1) and INMA(1) data, the estimated ? or ? will be close to zero, respectively. The results for all points in time are shown in Table ?8-43, Table ?8-44, and Table ?8-45. The results show that, for INARMA(0,0), identification produces better forecasts than the all-INARMA(1,1) approach. When the number of observations increases, the results of two approaches become close. For ?= 96, the degree of improvement by using identification rather than all-INARMA(1,1) is on average 2.3 percent. For INAR(1) and INMA(1) processes, when the number of observations is small, the all-INARMA(1,1) approach produces better results in many cases. But when the number of observations increases, the results of identification improve and the two methods produce close results. Based on the results of Table ?8-43, Table ?8-44, and Table ?8-45, using the most general model could be a good substitute for identification especially when less data is available as is often the case for intermittent demand data. Although it has not been looked at in the literature, the results suggest that treating the data as the general INARMA process and eliminating the complexity of identification, can be considered as a potentially useful approach. The forecast accuracy of all-INARMA(1,1) is compared to those of all-INAR(1) in Appendix 8.G. It is expected that for INARMA(0,0) and INAR(1) series the latter outperforms the former and the results confirm this for most of the cases. However, the results show that, even for INMA(1) and INARMA(1,1) series, and even for high moving average parameters, all-INAR(1) method produces more accurate forecasts (in terms of MSE and MASE) than all-INARMA(1,1) method in most of the cases. The difference increases for longer history. M.Mohammadipour, 2009, Chapter 8 189 Table 8-43 Accuracy of forecasts with identification and all-INARMA(1,1) for INARMA(0,0) series Parameters ?=?? ?=?? ?=?? ?=?? Two-stage identification All-INARMA(1,1) Two-stage identification All-INARMA(1,1) Two-stage identification All-INARMA(1,1) Two-stage identification All-INARMA(1,1) ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.3 0.0020 0.3496 1.1877 -0.0032 0.3692 1.2046 0.0019 0.3307 1.1525 -0.0048 0.3470 1.1612 0.0003 0.3190 1.0924 -0.0064 0.3374 1.1003 0.0028 0.3146 1.0229 -0.0032 0.3182 1.0161 ?= 0.5 -0.0010 0.5789 1.0799 -0.0088 0.6195 1.1038 0.0002 0.5493 1.0109 -0.0072 0.5774 1.0240 -0.0050 0.5209 0.9639 -0.0114 0.5404 0.9630 -0.0054 0.5099 0.9213 0.0011 0.5416 0.9403 ?= 0.7 0.0008 0.8074 0.9693 -0.0137 0.8690 0.9951 0.0011 0.7706 0.9111 -0.0057 0.8129 0.9260 -0.0023 0.7429 0.8764 -0.0193 0.7647 0.8797 -0.0022 0.7186 0.8378 -0.0010 0.7351 0.8524 ?= 1 -0.0001 1.1538 0.8665 -0.0203 1.2508 0.8980 0.0011 1.1076 0.8157 -0.0111 1.1551 0.8358 -0.0152 1.0705 0.7820 -0.0017 1.1163 0.8011 -0.0061 1.0289 0.7508 -0.0045 1.0540 0.7640 ?= 3 0.0042 3.4186 0.8175 -0.0324 3.5960 0.8346 0.0064 3.3024 0.7818 -0.0082 3.3893 0.7871 0.0162 3.2304 0.7654 -0.0151 3.3233 0.7691 -0.0025 3.0952 0.7386 -0.0081 3.1778 0.7455 ?= 5 0.0161 5.8873 0.8155 -0.0507 6.0384 0.8235 -0.0107 5.5208 0.7711 -0.0144 5.7687 0.7783 0.0042 5.4208 0.7613 0.0135 5.5495 0.7689 -0.0153 5.2324 0.7339 -0.0056 5.1751 0.7306 ?= 20 0.0509 22.9286 0.7872 -0.0146 24.9564 0.8225 -0.0421 21.9667 0.7702 -0.0514 22.8435 0.7773 -0.0156 21.5521 0.7400 -0.0269 22.0828 0.7551 -0.0252 20.5910 0.7322 -0.0400 21.1039 0.7334 Table 8-44 Accuracy of forecasts with identification and all-INARMA(1,1) for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? One-stage identification All-INARMA(1,1) One-stage identification All-INARMA(1,1) One-stage identification All-INARMA(1,1) One-stage identification All-INARMA(1,1) ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 -0.0005 0.6347 1.1208 -0.0008 0.6699 1.1275 0.0115 0.6260 1.1063 0.0049 0.6447 1.0710 0.0056 0.6005 1.0295 0.0111 0.6389 1.0340 0.0010 0.5826 0.9799 0.0010 0.5857 0.9845 ?= 0.5,?= 0.5 0.0381 1.0614 1.3967 0.0130 1.0056 1.3278 0.0050 0.9311 1.2226 0.0039 0.9436 1.2602 0.0064 0.8655 1.1425 0.0041 0.9131 1.1864 -0.0055 0.8014 1.0694 -0.0033 0.8186 1.0727 ?= 0.9,?= 0.5 0.0279 1.4243 1.4776 0.0347 1.2681 1.4405 0.0034 1.1610 1.2703 0.0344 1.1681 1.3181 -0.0103 1.0725 1.2157 0.0066 1.0919 1.2150 -0.0120 0.9981 1.1395 0.0206 1.0235 1.1707 ?= 0.1,?= 1 0.0063 1.3144 0.9415 -0.0152 1.3755 0.9455 -0.0024 1.2606 0.8758 -0.0028 1.2778 0.8819 0.0071 1.2258 0.8523 -0.0066 1.1947 0.8427 0.0044 1.1590 0.8104 0.0045 1.1875 0.8235 ?= 0.5,?= 1 0.0196 2.0526 1.1509 -0.0092 1.9908 1.1188 0.0117 1.8415 1.0550 0.0050 1.8491 1.0640 0.0093 1.7273 1.0262 -0.0046 1.7735 1.0251 0.0039 1.6017 0.9651 0.0052 1.6469 0.9944 ?= 0.9,?= 1 -0.0212 2.7024 1.2864 0.0544 2.4325 1.2304 0.0037 2.2686 1.1378 0.0378 2.3301 1.1748 -0.0043 2.1714 1.1203 0.0484 2.1853 1.1423 -0.0012 2.0015 1.0693 0.0312 2.0343 1.0747 ?= 0.1,?= 3 -0.0310 3.8137 0.8345 -0.0275 4.1292 0.8818 0.0042 3.7900 0.8337 -0.0177 3.7457 0.8236 -0.0041 3.6297 0.8083 -0.0144 3.6442 0.8115 -0.0184 3.4845 0.7845 -0.0222 3.4929 0.7828 ?= 0.5,?= 3 0.0256 6.0453 1.0472 0.0237 5.8020 1.0279 0.0170 5.5157 0.9955 -0.0291 5.4831 0.9848 -0.0103 5.1944 0.9556 -0.0027 5.2166 0.9710 0.0027 4.7714 0.9155 0.0268 4.9748 0.9319 ?= 0.9,?= 3 -0.0198 8.0490 1.2229 0.1097 7.3168 1.1422 0.0090 6.7598 1.1070 0.1077 6.9143 1.1046 0.0301 6.4818 1.0660 0.0627 6.5204 1.0742 0.0099 6.0402 1.0160 0.0612 6.1440 1.0290 ?= 0.1,?= 5 0.0440 6.7467 0.8632 0.0124 6.5975 0.8522 0.0022 6.1377 0.8190 0.0016 6.3370 0.8206 -0.0241 6.1470 0.8087 -0.0078 6.2455 0.8197 -0.0252 5.9063 0.7726 -0.0063 5.8121 0.7805 ?= 0.5,?= 5 -0.0116 9.9783 1.0548 0.0798 9.5203 1.0114 -0.0001 9.0103 0.9843 0.0050 8.8544 0.9603 -0.0479 8.4696 0.9420 0.0325 8.4735 0.9460 -0.0078 7.9000 0.9092 -0.0003 7.9638 0.9095 ?= 0.9,?= 5 0.1313 13.5067 1.1973 0.0773 12.2386 1.1385 0.0554 11.3171 1.0809 0.1196 11.3739 1.0820 0.0363 10.8754 1.0706 0.1021 10.9289 1.0679 -0.0036 10.0731 1.0171 0.0956 10.1452 1.0232 M.Mohammadipour, 2009, Chapter 8 190 Table 8-45 Accuracy of forecasts with identification and all-INARMA(1,1) for INMA(1) series Parameters ?=?? ?=?? ?=?? ?=?? One-stage identification All-INARMA(1,1) One-stage identification All-INARMA(1,1) One-stage identification All-INARMA(1,1) One-stage identification All-INARMA(1,1) ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 0.0006 0.6474 1.1537 -0.0056 0.6500 1.1831 -0.0042 0.6022 1.0458 -0.0038 0.6155 1.0589 0.0005 0.5732 0.9729 -0.0071 0.6070 1.0025 -0.0088 0.6604 0.7515 0.0063 0.5733 0.9809 ?= 0.5,?= 0.5 0.0210 0.8392 1.2970 0.0205 0.8681 1.3428 -0.0001 0.7935 1.1635 -0.0151 0.8047 1.1628 0.0013 0.7284 1.0496 0.0011 0.7815 1.1069 0.0401 0.9122 0.9183 -0.0038 0.7423 1.0637 ?= 0.9,?= 0.5 -0.0101 1.9689 1.1383 0.0430 1.0089 1.3941 0.0269 0.9813 1.2952 0.0352 0.9383 1.2649 0.0304 0.8882 1.1203 0.0122 0.9436 1.1917 0.1589 1.0747 1.0178 0.0250 0.9208 1.1448 ?= 0.1,?= 1 -0.0104 1.2574 0.8783 -0.0202 1.3447 0.9356 -0.0014 1.2389 0.8692 -0.0154 1.2458 0.8487 -0.0103 1.1444 0.8014 -0.0017 1.1956 0.8443 -0.0335 1.2159 0.8080 -0.0016 1.1525 0.8117 ?= 0.5,?= 1 0.0241 1.7225 1.0816 0.0060 1.6743 1.0374 -0.0065 1.5827 0.9817 -0.0128 1.5905 0.9873 -0.0020 1.4386 0.9239 0.0030 1.5879 0.9894 0.0177 1.5554 0.9547 -0.0077 1.4733 0.9413 ?= 0.9,?= 1 -0.0131 1.9568 1.1574 0.0145 1.9996 1.1350 0.0036 1.8299 1.0827 0.0180 1.9324 1.0960 0.0179 1.6855 0.9954 0.0165 1.8242 1.0545 0.0595 1.7732 1.0167 0.0184 1.7825 1.0406 ?= 0.1,?= 3 0.0170 3.9592 0.8653 -0.0153 3.8953 0.8642 -0.0217 3.7610 0.8249 0.0022 3.7206 0.8211 0.0052 3.4847 0.7859 0.0041 3.6216 0.7983 0.0006 3.5011 0.8056 -0.0215 3.5003 0.7873 ?= 0.5,?= 3 0.0143 5.2555 0.9867 -0.0306 4.8945 0.9557 -0.0237 4.8094 0.9361 -0.0137 4.7351 0.9278 -0.0040 4.2445 0.8681 -0.0013 4.6193 0.9143 -0.0040 4.2567 0.8749 0.0061 4.3870 0.8783 ?= 0.9,?= 3 -0.0283 6.0747 1.0769 0.0763 5.8039 1.0545 0.0190 5.4653 1.0106 0.0192 5.5915 1.0116 -0.0034 4.8673 0.9248 0.0118 5.2638 0.9726 -0.0028 4.8710 0.9265 0.0439 5.1879 0.9612 ?= 0.1,?= 5 -0.0388 6.5789 0.8436 0.0108 6.5867 0.8570 -0.0241 6.1703 0.8007 -0.0108 6.2327 0.8241 0.0168 5.7806 0.7811 -0.0045 6.0357 0.7906 0.0160 5.7818 0.7843 -0.0103 5.7236 0.7712 ?= 0.5,?= 5 -0.0841 8.5133 0.9644 -0.0335 8.3565 0.9718 -0.0184 7.8372 0.9231 0.0231 7.7461 0.9228 0.0028 7.1503 0.8623 0.0136 7.6924 0.8954 0.0027 7.1503 0.8628 0.0160 7.2509 0.8698 ?= 0.9,?= 5 -0.0552 9.9051 1.0576 0.0927 9.8826 1.0531 0.0005 8.9502 0.9893 0.0643 8.8871 0.9798 0.0135 7.9759 0.9193 0.0683 8.9059 0.9673 0.0135 7.9764 0.9194 0.0524 8.4545 0.9443 M.Mohammadipour, 2009, Chapter 8 191 8.6.2.5 All-INAR(1) vs Benchmark Methods Based on the argument in the previous section, the degree of improvement by treating all INARMA series as an INAR(1) model over the benchmark methods is investigated in this section. The MSE results for INARMA(0,0), INMA(1), and INARMA(1,1) series are shown in Table 8-46 to Table ?8-49. The results for INAR(1) series are the same as the results of known order (Table ?8-18 and Table ?8-19). The results are for the case that all points in time are considered. As previously mentioned in section 8.6.1, there was a slight improvement over the benchmark methods when demand is INARMA(0,0) or INMA(1). This was for the case that the order of the INARMA model was known. Considering the fact that the identification errors result in deterioration of forecasting accuracy for INARMA models, we except benchmark methods to outperform INARMA especially for more sparse demand. The results of Table 8-46 and Table ?8-47 confirm this. Table 8-46 for INARMA(0,0) series (unknown order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.8998 0.9435 0.9473 0.9746 0.9991 1.0012 1.0070 1.0230 1.0239 0.9918 0.9997 1.0000 ?= 0.5 0.9838 1.0119 1.0136 0.9998 1.0150 1.0159 1.0045 1.0157 1.0163 0.9756 0.9850 0.9854 ?= 0.7 1.0048 1.0262 1.0280 0.9922 1.0049 1.0054 0.9845 0.9952 0.9955 0.9663 0.9764 0.9767 ?= 1 1.0031 1.0179 1.0181 0.9785 0.9887 0.9885 0.9440 0.9533 0.9531 0.9624 0.9719 0.9716 ?= 3 1.0170 1.0095 1.0051 0.9479 0.9424 0.9385 0.9496 0.9428 0.9388 0.9198 0.9126 0.9087 ?= 5 0.9647 0.9399 0.9321 0.9383 0.9128 0.9050 0.9593 0.9364 0.9288 0.9290 0.9048 0.8972 ?= 20 0.9677 0.8339 0.8063 0.9323 0.8030 0.7762 0.9686 0.8339 0.8058 0.9276 0.7982 0.7712 Table 8-47 for INMA(1) series (unknown order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 1.0350 1.0693 1.0717 0.9922 1.0109 1.0122 0.9695 0.9813 0.9818 0.9558 0.9678 0.9683 ?= 0.5,?= 0.5 0.8597 0.8923 0.8941 0.8599 0.8828 0.8842 0.8933 0.9129 0.9141 0.8192 0.8369 0.8380 ?= 0.9,?= 0.5 0.7818 0.8177 0.8200 0.7585 0.7812 0.7826 0.7387 0.7571 0.7580 0.7201 0.7387 0.7398 ?= 0.1,?= 1 1.0022 1.0157 1.0157 0.9694 0.9819 0.9818 0.9695 0.9804 0.9801 0.9598 0.9705 0.9703 ?= 0.5,?= 1 0.9325 0.9471 0.9468 0.9167 0.9269 0.9261 0.8776 0.8891 0.8885 0.8698 0.8809 0.8803 ?= 0.9,?= 1 0.8373 0.8436 0.8421 0.7809 0.7884 0.7873 0.7796 0.7868 0.7856 0.7468 0.7528 0.7515 ?= 0.1,?= 3 0.9677 0.9569 0.9521 0.9527 0.9430 0.9383 0.9385 0.9276 0.9228 0.9381 0.9295 0.9250 ?= 0.5,?= 3 0.9425 0.9215 0.9143 0.9388 0.9173 0.9101 0.9063 0.8860 0.8791 0.8884 0.8678 0.8610 ?= 0.9,?= 3 0.8611 0.8259 0.8166 0.8248 0.7942 0.7855 0.8116 0.7835 0.7752 0.8002 0.7705 0.7621 ?= 0.1,?= 5 0.9625 0.9316 0.9227 0.9444 0.9198 0.9116 0.9639 0.9356 0.9269 0.9229 0.8952 0.8868 ?= 0.5,?= 5 0.9521 0.9031 0.8903 0.9356 0.8843 0.8713 0.9162 0.8651 0.8525 0.8655 0.8224 0.8108 ?= 0.9,?= 5 0.8754 0.8149 0.8001 0.8617 0.8037 0.7889 0.8093 0.7546 0.7409 0.7908 0.7355 0.7219 BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 192 The results for INARMA(1,1) series again show that with the presence of a high autoregressive parameter, INARMA has a considerably smaller MSE than the benchmark methods. The results of MASE also confirm this (see Appendix 8.H). Table 8-48 with smoothing parameter 0.2 for INARMA(1,1) series (unknown order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9452 0.9745 0.9759 0.9301 0.9500 0.9513 0.9733 0.9899 0.9910 0.9297 0.9444 0.9451 ?= 0.1,?= 0.9,?= 0.5 0.7664 0.7989 0.8007 0.7325 0.7564 0.7581 0.7616 0.7811 0.7821 0.7025 0.7196 0.7206 ?= 0.5,?= 0.5,?= 0.5 0.6855 0.7039 0.7039 0.6828 0.6985 0.6990 0.6243 0.6358 0.6359 0.6193 0.6332 0.6336 ?= 0.9,?= 0.1,?= 0.5 0.5274 0.4433 0.4294 0.5164 0.4542 0.4414 0.4968 0.4385 0.4264 0.4740 0.4208 0.4091 ?= 0.1,?= 0.1,?= 1 0.9771 0.9922 0.9922 0.9486 0.9621 0.9620 0.9571 0.9695 0.9693 0.9168 0.9284 0.9282 ?= 0.1,?= 0.9,?= 1 0.8802 0.8867 0.8851 0.7635 0.7730 0.7721 0.7569 0.7619 0.7605 0.7493 0.7556 0.7544 ?= 0.5,?= 0.5,?= 1 0.7720 0.7663 0.7629 0.7394 0.7370 0.7340 0.6899 0.6858 0.6828 0.6710 0.6701 0.6675 ?= 0.9,?= 0.1,?= 1 0.5481 0.4206 0.3994 0.5058 0.3942 0.3750 0.4906 0.3856 0.3671 0.4525 0.3597 0.3428 ?= 0.1,?= 0.1,?= 5 0.9920 0.9531 0.9426 0.9574 0.9233 0.9135 0.9436 0.9097 0.8999 0.9182 0.8846 0.8750 ?= 0.1,?= 0.9,?= 5 0.9232 0.8501 0.8332 0.8544 0.7861 0.7705 0.8255 0.7633 0.7485 0.7985 0.7393 0.7249 ?= 0.5,?= 0.5,?= 5 0.8379 0.7406 0.7199 0.7689 0.6783 0.6591 0.7251 0.6455 0.6280 0.7068 0.6278 0.6106 ?= 0.9,?= 0.1,?= 5 0.5482 0.2488 0.2197 0.5407 0.2289 0.2017 0.4987 0.2170 0.1916 0.4535 0.2013 0.1779 Table 8-49 with smoothing parameter 0.5 for INARMA(1,1) series (unknown order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.8646 0.9584 0.9809 0.8208 0.9087 0.9414 0.8490 0.9381 0.9726 0.8068 0.8896 0.9256 ?= 0.1,?= 0.9,?= 0.5 0.7087 0.7961 0.7972 0.6459 0.7323 0.7446 0.6773 0.7553 0.7557 0.6244 0.6964 0.6958 ?= 0.5,?= 0.5,?= 0.5 0.7141 0.7652 0.6628 0.7106 0.7497 0.6526 0.6482 0.6801 0.5903 0.6404 0.6781 0.5906 ?= 0.9,?= 0.1,?= 0.5 0.8766 0.3474 0.1933 0.8457 0.3490 0.2002 0.8296 0.3417 0.1954 0.7896 0.3213 0.1843 ?= 0.1,?= 0.1,?= 1 0.8757 0.9304 0.9275 0.8436 0.9003 0.8991 0.8544 0.9068 0.9031 0.8140 0.8656 0.8655 ?= 0.1,?= 0.9,?= 1 0.8592 0.8631 0.7723 0.7282 0.7454 0.6831 0.7346 0.7378 0.6648 0.7217 0.7292 0.6608 ?= 0.5,?= 0.5,?= 1 0.8696 0.7883 0.6142 0.8291 0.7581 0.5942 0.7723 0.7027 0.5508 0.7463 0.6873 0.5413 ?= 0.9,?= 0.1,?= 1 0.8825 0.2253 0.1286 0.8418 0.2177 0.1234 0.8206 0.2131 0.1211 0.7720 0.2016 0.1143 ?= 0.1,?= 0.1,?= 5 0.8766 0.7247 0.6268 0.8494 0.7041 0.6099 0.8431 0.6967 0.5997 0.8140 0.6748 0.5838 ?= 0.1,?= 0.9,?= 5 0.9239 0.6033 0.4448 0.8616 0.5584 0.4126 0.8308 0.5450 0.4037 0.8036 0.5254 0.3890 ?= 0.5,?= 0.5,?= 5 0.9620 0.4915 0.3263 0.8759 0.4476 0.2977 0.8274 0.4318 0.2883 0.8025 0.4159 0.2780 ?= 0.9,?= 0.1,?= 5 0.9115 0.0647 0.0366 0.8855 0.0596 0.0336 0.8317 0.0572 0.0323 0.7628 0.0531 0.0300 BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Chapter 8 193 8.6.3 Lead Time Forecasts In chapter 6, the lead time forecasts for the INARMA models were presented. For INARMA(0,0) and INMA(1) processes, the lead time forecast is simply given by: ??? ??+? ?+1 ?=1 |???= (?+ 1)? Equation 8-18 ??? ??+? ?+1 ?=1 |???=??+ 1?(1 +?)? Equation 8-19 For INAR(1) and INARMA(1,1) processes, the lead time forecasts are: ??? ??+? ?+1 ?=1 |???= ?(1???+1) 1?? ??+ ? 1?? ???+ 1??? ?? ?+1 ?=1 ? Equation 8-20 ??? ??+? ?+1 ?=1 |???= ?(1???+1) 1?? ??+ ?(1 +?) 1?? ???+ 1??? ?? ?+1 ?=1 ? Equation 8-21 The results of sections 8.6.2.4 and 8.6.2.5 show that the accuracy of forecasts of an all-INAR(1) method are generally better than those of an all-INARMA(1,1) method even for INARMA(1,1) series. Therefore, in this section we use an all-INAR(1) method and compare the lead time forecasts of this method with those of benchmarks. The results of comparing the MSE of INARMA with that of benchmark methods for INARMA(0,0), INMA(1), INAR(1) and INARMA(1,1) series are presented in Table ?8-50 to Table ?8-61. This includes both cases of ?= 3 and ?= 6. The results using MASE are presented in Appendix 8.I. For INARMA(0,0) series, the results of Table ?8-50 show that the all-INAR(1) lead time forecasts are better than the best benchmark in most of the cases (with an exception of ?= 0.7,?= 24). The same is true for INMA(1) series, with some exceptions for ?= 24. M.Mohammadipour, 2009, Chapter 8 194 Table 8-50 BenchmarkINARMA MSEMSE / of lead-time forecasts for INARMA(0,0) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.7373 0.8307 0.8409 0.8726 0.9393 0.9457 0.9482 0.9880 0.9912 0.9448 0.9681 0.9695 ?= 0.5 0.9128 0.9879 0.9946 0.9573 0.9968 0.9992 0.9467 0.9804 0.9822 0.9097 0.9357 0.9367 ?= 0.7 0.9583 1.0130 1.0166 0.9541 0.9862 0.9872 0.9332 0.9606 0.9611 0.8914 0.9142 0.9143 ?= 1 0.9589 0.9878 0.9876 0.9361 0.9625 0.9620 0.9040 0.9265 0.9257 0.8601 0.8835 0.8830 ?= 3 0.9238 0.9077 0.8984 0.8642 0.8432 0.8338 0.8292 0.8141 0.8055 0.8021 0.7884 0.7803 ?= 5 0.9120 0.8519 0.8342 0.8657 0.8127 0.7967 0.8270 0.7755 0.7598 0.7897 0.7418 0.7272 ?= 20 0.9025 0.6417 0.5990 0.8521 0.6085 0.5675 0.8197 0.5886 0.5494 0.7793 0.5561 0.5189 Table 8-51 of lead-time forecasts for INMA(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9226 1.0030 1.0104 0.9503 0.9928 0.9956 0.9429 0.9749 0.9765 0.9075 0.9372 0.9386 ?= 0.5,?= 0.5 0.9668 1.0400 1.0463 0.9292 0.9766 0.9800 0.9077 0.9468 0.9493 0.8554 0.8898 0.8918 ?= 0.9,?= 0.5 0.9745 1.0630 1.0708 0.9331 0.9877 0.9916 0.9030 0.9444 0.9469 0.8419 0.8803 0.8826 ?= 0.1,?= 1 0.9935 1.0339 1.0348 0.9341 0.9628 0.9627 0.9005 0.9262 0.9258 0.8557 0.8814 0.8812 ?= 0.5,?= 1 1.0054 1.0366 1.0366 0.9199 0.9444 0.9438 0.8925 0.9156 0.9148 0.8340 0.8579 0.8575 ?= 0.9,?= 1 1.0143 1.0469 1.0465 0.9382 0.9631 0.9621 0.9013 0.9218 0.9203 0.8360 0.8568 0.8557 ?= 0.1,?= 3 0.9407 0.9237 0.9139 0.8797 0.8710 0.8626 0.8487 0.8312 0.8223 0.8107 0.7981 0.7900 ?= 0.5,?= 3 0.9729 0.9451 0.9334 0.9029 0.8737 0.8627 0.8729 0.8508 0.8408 0.8190 0.7970 0.7873 ?= 0.9,?= 3 1.0083 0.9679 0.9541 0.9234 0.8801 0.8666 0.8841 0.8521 0.8401 0.8274 0.7961 0.7846 ?= 0.1,?= 5 0.9367 0.8801 0.8631 0.8710 0.8150 0.7988 0.8364 0.7885 0.7732 0.8075 0.7592 0.7443 ?= 0.5,?= 5 0.9695 0.9048 0.8853 0.8995 0.8289 0.8103 0.8683 0.8063 0.7887 0.8225 0.7612 0.7442 ?= 0.9,?= 5 0.9940 0.9124 0.8899 0.9071 0.8238 0.8029 0.8743 0.7985 0.7784 0.8297 0.7582 0.7392 Table 8-52 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9441 1.0142 1.0203 0.9422 0.9906 0.9942 0.9398 0.9719 0.9736 0.9012 0.9277 0.9289 ?= 0.5,?= 0.5 0.9037 0.9525 0.9560 0.8704 0.9106 0.9131 0.8268 0.8611 0.8631 0.7714 0.8063 0.8085 ?= 0.9,?= 0.5 0.8049 0.6984 0.6780 0.7157 0.6624 0.6473 0.6897 0.6391 0.6243 0.6219 0.5734 0.5605 ?= 0.1,?= 1 0.9950 1.0303 1.0307 0.9124 0.9404 0.9403 0.9040 0.9314 0.9313 0.8533 0.8793 0.8792 ?= 0.5,?= 1 0.9647 0.9843 0.9824 0.8941 0.9126 0.9110 0.8683 0.8820 0.8799 0.8117 0.8272 0.8254 ?= 0.9,?= 1 0.7938 0.6568 0.6283 0.7219 0.6001 0.5749 0.6752 0.5758 0.5528 0.6244 0.5242 0.5024 ?= 0.1,?= 3 0.9404 0.9214 0.9117 0.8876 0.8741 0.8649 0.8494 0.8370 0.8286 0.8052 0.7906 0.7825 ?= 0.5,?= 3 0.9868 0.9416 0.9267 0.9043 0.8639 0.8504 0.8770 0.8385 0.8255 0.8203 0.7864 0.7745 ?= 0.9,?= 3 0.7825 0.4666 0.4243 0.7136 0.4197 0.3827 0.6830 0.4084 0.3724 0.6364 0.3850 0.3508 ?= 0.1,?= 5 0.9398 0.8872 0.8696 0.8577 0.8118 0.7964 0.8487 0.7989 0.7833 0.8011 0.7536 0.7389 ?= 0.5,?= 5 0.9849 0.8914 0.8674 0.9053 0.8171 0.7949 0.8794 0.8054 0.7844 0.8246 0.7455 0.7253 ?= 0.9,?= 5 0.7978 0.3639 0.3223 0.7284 0.3349 0.2968 0.6803 0.3201 0.2843 0.6294 0.3002 0.2666 )( 3?l BenchmarkINARMA MSEMSE / )( 3?l BenchmarkINARMA MSEMSE / )( 3?l M.Mohammadipour, 2009, Chapter 8 195 Table 8-53 BenchmarkINARMA MSEMSE / of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.8202 0.9889 1.0012 0.7103 0.8761 0.8946 0.7140 0.8540 0.8617 0.6728 0.8053 0.8140 ?= 0.5,?= 0.5 0.8152 0.9551 0.9479 0.7505 0.8866 0.8825 0.7104 0.8328 0.8258 0.6566 0.7783 0.7749 ?= 0.9,?= 0.5 1.0550 0.5086 0.3537 0.9908 0.5117 0.3575 0.9417 0.4827 0.3368 0.8686 0.4493 0.3156 ?= 0.1,?= 1 0.7599 0.8681 0.8403 0.6779 0.7799 0.7567 0.6688 0.7718 0.7501 0.6283 0.7240 0.7033 ?= 0.5,?= 1 0.8607 0.8891 0.8204 0.7999 0.8315 0.7695 0.7669 0.7907 0.7273 0.7220 0.7482 0.6894 ?= 0.9,?= 1 1.0446 0.3595 0.2321 0.9967 0.3383 0.2190 0.9427 0.3302 0.2135 0.8687 0.2953 0.1905 ?= 0.1,?= 3 0.6633 0.6147 0.5303 0.6263 0.5800 0.4968 0.6038 0.5614 0.4833 0.5732 0.5315 0.4577 ?= 0.5,?= 3 0.8636 0.6624 0.5257 0.7882 0.6057 0.4822 0.7691 0.5909 0.4702 0.7130 0.5554 0.4434 ?= 0.9,?= 3 1.0440 0.1492 0.0887 0.9923 0.1413 0.0843 0.9500 0.1361 0.0812 0.8741 0.1263 0.0751 ?= 0.1,?= 5 0.6527 0.5102 0.4046 0.5950 0.4720 0.3774 0.5958 0.4671 0.3722 0.5596 0.4401 0.3514 ?= 0.5,?= 5 0.8592 0.5293 0.3882 0.7813 0.4819 0.3539 0.7587 0.4735 0.3476 0.7183 0.4396 0.3219 ?= 0.9,?= 5 1.0677 0.0967 0.0564 0.9947 0.0895 0.0522 0.9433 0.0866 0.0505 0.8708 0.0809 0.0472 The results of Table ?8-53 for INAR(1) series reveal that, for high autoregressive parameters, the improvement of INARMA over the best benchmark is narrow for short length of history (in case of ?= 24, INARMA is even worse). However with more observations, the improvement also increases. For small autoregressive parameters, the results of Table ?8-52 show that INARMA always outperforms the benchmark methods. Again, the improvement increases with an increase in the length of history. Table 8-54 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9369 1.0169 1.0242 0.9479 0.9921 0.9951 0.9330 0.9650 0.9668 0.8867 0.9178 0.9195 ?= 0.1,?= 0.9,?= 0.5 1.0164 1.0937 1.1002 0.9385 0.9915 0.9953 0.8892 0.9311 0.9338 0.8316 0.8679 0.8700 ?= 0.5,?= 0.5,?= 0.5 0.9728 1.0101 1.0118 0.8611 0.8924 0.8939 0.8120 0.8471 0.8490 0.7593 0.7894 0.7908 ?= 0.9,?= 0.1,?= 0.5 0.7710 0.6826 0.6646 0.7032 0.6297 0.6135 0.6573 0.6016 0.5875 0.6034 0.5641 0.5519 ?= 0.1,?= 0.1,?= 1 0.9975 1.0335 1.0342 0.9151 0.9411 0.9409 0.9053 0.9308 0.9305 0.8528 0.8780 0.8779 ?= 0.1,?= 0.9,?= 1 1.0291 1.0487 1.0470 0.9294 0.9539 0.9530 0.8891 0.9065 0.9048 0.8321 0.8514 0.8502 ?= 0.5,?= 0.5,?= 1 0.9776 0.9953 0.9927 0.8819 0.8938 0.8909 0.8278 0.8396 0.8372 0.7817 0.7921 0.7897 ?= 0.9,?= 0.1,?= 1 0.7905 0.6207 0.5913 0.7148 0.5925 0.5671 0.6650 0.5543 0.5310 0.6202 0.5197 0.4979 ?= 0.1,?= 0.1,?= 5 0.9627 0.9084 0.8908 0.8969 0.8410 0.8240 0.8528 0.7949 0.7788 0.8188 0.7691 0.7541 ?= 0.1,?= 0.9,?= 5 0.9897 0.9024 0.8794 0.9067 0.8267 0.8062 0.8628 0.7857 0.7659 0.8240 0.7549 0.7362 ?= 0.5,?= 0.5,?= 5 0.9816 0.8746 0.8481 0.8829 0.7884 0.7650 0.8442 0.7589 0.7369 0.7974 0.7113 0.6900 ?= 0.9,?= 0.1,?= 5 0.8126 0.3508 0.3100 0.7160 0.3352 0.2973 0.6811 0.3100 0.2747 0.6252 0.2905 0.2576 For INARMA(1,1) series, the results of Table ?8-54 reveal that for small autoregressive parameters, INARMA outperforms the benchmark methods in most cases (except for )( 3?l BenchmarkINARMA MSEMSE / )( 3?l M.Mohammadipour, 2009, Chapter 8 196 sparse data and short history). The improvement increases with an increase in the length of history. For high autoregressive parameters, the improvement of INARMA over the best benchmark is narrow for short length of history (in case of ?= 24, INARMA is even worse). However, with more observations, the improvement also increases. Table 8-55 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.7926 0.9844 1.0046 0.7387 0.8943 0.9066 0.7080 0.8508 0.8602 0.6609 0.8020 0.8151 ?= 0.1,?= 0.9,?= 0.5 0.8284 1.0145 1.0283 0.7348 0.8943 0.9020 0.7039 0.8473 0.8523 0.6502 0.7844 0.7891 ?= 0.5,?= 0.5,?= 0.5 0.8944 1.0074 0.9783 0.7886 0.8929 0.8719 0.7309 0.8355 0.8170 0.6878 0.7789 0.7584 ?= 0.9,?= 0.1,?= 0.5 1.0603 0.5206 0.3644 0.9695 0.4808 0.3352 0.9265 0.4691 0.3282 0.8538 0.4430 0.3099 ?= 0.1,?= 0.1,?= 1 0.7832 0.8878 0.8586 0.6954 0.7909 0.7656 0.6774 0.7742 0.7505 0.6433 0.7359 0.7132 ?= 0.1,?= 0.9,?= 1 0.8336 0.8953 0.8471 0.7351 0.7975 0.7544 0.7160 0.7657 0.7200 0.6662 0.7168 0.6756 ?= 0.5,?= 0.5,?= 1 0.9017 0.9086 0.8233 0.8102 0.8075 0.7288 0.7721 0.7688 0.6955 0.7151 0.7143 0.6467 ?= 0.9,?= 0.1,?= 1 1.0454 0.3365 0.2171 0.9817 0.3287 0.2121 0.9322 0.3129 0.2021 0.8667 0.2929 0.1888 ?= 0.1,?= 0.1,?= 5 0.6847 0.5329 0.4230 0.6441 0.4943 0.3906 0.6123 0.4713 0.3743 0.5900 0.4578 0.3637 ?= 0.1,?= 0.9,?= 5 0.7799 0.5191 0.3907 0.7282 0.4872 0.3678 0.6804 0.4559 0.3444 0.6462 0.4364 0.3297 ?= 0.5,?= 0.5,?= 5 0.8976 0.5027 0.3589 0.8053 0.4581 0.3287 0.7748 0.4442 0.3189 0.7296 0.4119 0.2948 ?= 0.9,?= 0.1,?= 5 1.0675 0.0921 0.0538 1.0085 0.0898 0.0523 0.9440 0.0829 0.0483 0.8732 0.0781 0.0455 When ?= 6, the results of Table ?8-56 and Table ?8-57 show that the improvement by using INARMA (an all-INAR(1) method) over the benchmarks is generally greater than the case of ?= 3 for INARMA(0,0) and INMA(1) series. This is also true for INAR(1) and INARMA(1,1) series with small autoregressive parameters. Table 8-56 of lead-time forecasts for INARMA(0,0) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.7796 0.9163 0.9315 0.7748 0.8717 0.8815 0.9112 0.9772 0.9824 0.9101 0.9488 0.9509 ?= 0.5 0.8894 1.0054 1.0161 0.8839 0.9565 0.9615 0.8798 0.9363 0.9394 0.8343 0.8782 0.8801 ?= 0.7 0.9194 1.0260 1.0341 0.9137 0.9761 0.9785 0.8802 0.9208 0.9212 0.7996 0.8416 0.8424 ?= 1 0.9390 1.0161 1.0193 0.8674 0.9197 0.9202 0.8225 0.8631 0.8623 0.7647 0.7959 0.7945 ?= 3 0.8688 0.8472 0.8330 0.7845 0.7530 0.7397 0.7503 0.7332 0.7213 0.6768 0.6616 0.6507 ?= 5 0.8601 0.7770 0.7533 0.7800 0.6999 0.6783 0.7253 0.6539 0.6336 0.6543 0.5956 0.5778 ?= 20 0.8767 0.5377 0.4903 0.7646 0.4681 0.4264 0.7239 0.4344 0.3956 0.6602 0.3997 0.3638 BenchmarkINARMA MSEMSE / )( 3?l BenchmarkINARMA MSEMSE / )( 6?l M.Mohammadipour, 2009, Chapter 8 197 Table 8-57 of lead-time forecasts for INMA(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9555 1.0866 1.0992 0.9010 0.9718 0.9768 0.8826 0.9284 0.9306 0.8254 0.8694 0.8715 ?= 0.5,?= 0.5 0.9758 1.1192 1.1334 0.9130 0.9811 0.9859 0.8869 0.9499 0.9540 0.7785 0.8321 0.8355 ?= 0.9,?= 0.5 0.9807 1.1343 1.1492 0.9248 1.0144 1.0212 0.8610 0.9261 0.9304 0.7866 0.8389 0.8420 ?= 0.1,?= 1 0.9389 0.9944 0.9952 0.8628 0.9081 0.9082 0.8151 0.8529 0.8524 0.7649 0.7994 0.7987 ?= 0.5,?= 1 1.0045 1.0576 1.0578 0.8718 0.9157 0.9156 0.8293 0.8652 0.8644 0.7372 0.7696 0.7690 ?= 0.9,?= 1 1.0814 1.1128 1.1094 0.8983 0.9348 0.9332 0.8351 0.8787 0.8780 0.7429 0.7737 0.7725 ?= 0.1,?= 3 0.9124 0.8748 0.8592 0.8079 0.7739 0.7602 0.7394 0.7149 0.7030 0.6741 0.6553 0.6447 ?= 0.5,?= 3 0.9662 0.9205 0.9032 0.8329 0.8094 0.7957 0.7654 0.7379 0.7251 0.7011 0.6748 0.6630 ?= 0.9,?= 3 1.0009 0.9633 0.9456 0.8596 0.8297 0.8139 0.7917 0.7518 0.7364 0.7021 0.6679 0.6544 ?= 0.1,?= 5 0.9127 0.8509 0.8281 0.7848 0.7108 0.6897 0.7241 0.6617 0.6424 0.6698 0.6068 0.5887 ?= 0.5,?= 5 0.9715 0.8770 0.8489 0.8084 0.7152 0.6921 0.7664 0.6978 0.6771 0.6977 0.6281 0.6085 ?= 0.9,?= 5 0.9922 0.8905 0.8600 0.8249 0.7123 0.6872 0.7697 0.6726 0.6487 0.7043 0.6223 0.6010 Table 8-58 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.8625 0.9847 0.9967 0.9076 0.9769 0.9817 0.8948 0.9484 0.9515 0.8221 0.8695 0.8720 ?= 0.5,?= 0.5 1.0555 1.1742 1.1849 0.9302 0.9968 1.0014 0.8466 0.9094 0.9138 0.7524 0.8044 0.8079 ?= 0.9,?= 0.5 1.0075 0.9778 0.9611 0.8818 0.8539 0.8397 0.8450 0.8217 0.8080 0.7595 0.7278 0.7149 ?= 0.1,?= 1 0.9882 1.0425 1.0431 0.8784 0.9175 0.9170 0.8314 0.8690 0.8684 0.7560 0.7902 0.7896 ?= 0.5,?= 1 1.0573 1.0997 1.0987 0.9195 0.9563 0.9552 0.8406 0.8812 0.8809 0.7540 0.7849 0.7841 ?= 0.9,?= 1 1.0224 0.8642 0.8311 0.8859 0.7750 0.7469 0.8428 0.7480 0.7224 0.7515 0.6479 0.6238 ?= 0.1,?= 3 0.9045 0.8885 0.8752 0.8091 0.7854 0.7727 0.7458 0.7196 0.7073 0.6899 0.6730 0.6625 ?= 0.5,?= 3 1.0213 0.9597 0.9405 0.8731 0.8446 0.8301 0.8273 0.8022 0.7887 0.7472 0.7167 0.7039 ?= 0.9,?= 3 1.0113 0.6310 0.5780 0.8916 0.5759 0.5289 0.8439 0.5512 0.5061 0.7476 0.4875 0.4483 ?= 0.1,?= 5 0.9003 0.8337 0.8110 0.7897 0.7115 0.6909 0.7370 0.6712 0.6516 0.6750 0.6152 0.5977 ?= 0.5,?= 5 1.0195 0.9360 0.9091 0.8660 0.7777 0.7532 0.8137 0.7304 0.7077 0.7336 0.6596 0.6391 ?= 0.9,?= 5 1.0231 0.5077 0.4525 0.8918 0.4638 0.4151 0.8473 0.4276 0.3828 0.7593 0.3885 0.3471 Table 8-59 of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.7025 0.9582 0.9857 0.6247 0.8199 0.8331 0.5745 0.7630 0.7799 0.5215 0.6905 0.7043 ?= 0.5,?= 0.5 0.8524 1.1231 1.1434 0.7197 0.9131 0.9156 0.6479 0.8299 0.8336 0.5690 0.7232 0.7255 ?= 0.9,?= 0.5 1.1450 0.7467 0.5466 1.0414 0.6744 0.4948 1.0005 0.6378 0.4654 0.8948 0.5661 0.4149 ?= 0.1,?= 1 0.6911 0.8284 0.7860 0.5767 0.6950 0.6643 0.5338 0.6439 0.6137 0.4794 0.5823 0.5562 ?= 0.5,?= 1 0.8340 0.9221 0.8532 0.7122 0.7868 0.7257 0.6494 0.7205 0.6662 0.5808 0.6405 0.5915 ?= 0.9,?= 1 1.1542 0.4903 0.3295 1.0352 0.4408 0.2939 0.9963 0.4401 0.2945 0.8878 0.3718 0.2478 ?= 0.1,?= 3 0.5518 0.5061 0.4185 0.4869 0.4383 0.3619 0.4381 0.3939 0.3249 0.4151 0.3808 0.3157 ?= 0.5,?= 3 0.7784 0.5975 0.4667 0.6520 0.5253 0.4146 0.6223 0.4970 0.3921 0.5562 0.4413 0.3479 ?= 0.9,?= 3 1.1314 0.2106 0.1279 1.0516 0.1967 0.1188 0.9973 0.1881 0.1133 0.8818 0.1691 0.1023 ?= 0.1,?= 5 0.5334 0.4028 0.3086 0.4696 0.3467 0.2660 0.4351 0.3195 0.2432 0.4036 0.2995 0.2290 ?= 0.5,?= 5 0.7723 0.4940 0.3607 0.6419 0.4011 0.2909 0.6079 0.3820 0.2778 0.5447 0.3420 0.2489 ?= 0.9,?= 5 1.1593 0.1391 0.0817 1.0545 0.1297 0.0765 0.9927 0.1220 0.0720 0.8994 0.1079 0.0635 BenchmarkINARMA MSEMSE / )( 6?l BenchmarkINARMA MSEMSE / )( 6?l BenchmarkINARMA MSEMSE / )( 6?l M.Mohammadipour, 2009, Chapter 8 198 Table 8-60 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9060 1.0221 1.0332 0.9114 0.9817 0.9867 0.8773 0.9416 0.9460 0.8151 0.8622 0.8649 ?= 0.1,?= 0.9,?= 0.5 1.1265 1.2635 1.2754 0.9392 1.0143 1.0196 0.8486 0.9209 0.9263 0.7638 0.8196 0.8232 ?= 0.5,?= 0.5,?= 0.5 1.1516 1.2325 1.2381 0.9878 1.0498 1.0535 0.8733 0.9306 0.9341 0.7564 0.8034 0.8062 ?= 0.9,?= 0.1,?= 0.5 1.0244 0.9352 0.9148 0.9012 0.8209 0.8031 0.8309 0.7946 0.7805 0.7339 0.7067 0.6946 ?= 0.1,?= 0.1,?= 1 0.9968 1.0472 1.0475 0.8069 0.7319 0.7102 0.8184 0.8626 0.8631 0.7493 0.7859 0.7857 ?= 0.1,?= 0.9,?= 1 1.0785 1.1185 1.1166 0.9066 0.9557 0.9558 0.8358 0.8781 0.8777 0.7402 0.7684 0.7670 ?= 0.5,?= 0.5,?= 1 1.1171 1.1825 1.1829 0.9376 0.9673 0.9650 0.8410 0.8700 0.8683 0.7467 0.7730 0.7715 ?= 0.9,?= 0.1,?= 1 1.0111 0.8583 0.8239 0.9020 0.7835 0.7542 0.8249 0.7473 0.7226 0.7269 0.6397 0.6174 ?= 0.1,?= 0.1,?= 5 0.9241 0.8565 0.8332 0.8016 0.7424 0.7218 0.7275 0.6645 0.6456 0.6887 0.6289 0.6111 ?= 0.1,?= 0.9,?= 5 0.9912 0.8566 0.8262 0.8560 0.7602 0.7341 0.7844 0.7027 0.6792 0.7024 0.6251 0.6044 ?= 0.5,?= 0.5,?= 5 1.0538 0.9423 0.9107 0.9001 0.7968 0.7694 0.8238 0.7342 0.7098 0.7444 0.6602 0.6380 ?= 0.9,?= 0.1,?= 5 1.0067 0.4899 0.4369 0.8862 0.4498 0.4017 0.8352 0.4233 0.3787 0.7291 0.3757 0.3360 Table 8-61 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.7042 0.9695 1.0084 0.6267 0.8313 0.8481 0.5555 0.7524 0.7762 0.5135 0.6838 0.7022 ?= 0.1,?= 0.9,?= 0.5 0.8479 1.1332 1.1494 0.6814 0.8807 0.8870 0.5838 0.7769 0.7926 0.5222 0.6861 0.6938 ?= 0.5,?= 0.5,?= 0.5 0.9535 1.1744 1.1651 0.8018 0.9757 0.9598 0.6933 0.8499 0.8383 0.6112 0.7413 0.7295 ?= 0.9,?= 0.1,?= 0.5 1.1333 0.7081 0.5236 1.0656 0.6303 0.4630 0.9797 0.6255 0.4586 0.8789 0.5565 0.4072 ?= 0.1,?= 0.1,?= 1 0.6951 0.8151 0.7736 0.4846 0.3506 0.2659 0.5338 0.6524 0.6276 0.4804 0.5845 0.5592 ?= 0.1,?= 0.9,?= 1 0.7753 0.8666 0.8034 0.6309 0.7326 0.6886 0.5860 0.6681 0.6238 0.5191 0.5816 0.5393 ?= 0.5,?= 0.5,?= 1 0.8907 0.9756 0.8926 0.7496 0.7941 0.7204 0.6635 0.7086 0.6475 0.5940 0.6314 0.5747 ?= 0.9,?= 0.1,?= 1 1.1266 0.4673 0.3099 1.0550 0.4426 0.2935 0.9801 0.4372 0.2912 0.8728 0.3766 0.2513 ?= 0.1,?= 0.1,?= 5 0.5502 0.4146 0.3174 0.4870 0.3610 0.2743 0.4375 0.3241 0.2474 0.4218 0.3097 0.2358 ?= 0.1,?= 0.9,?= 5 0.6697 0.4177 0.3047 0.5673 0.3623 0.2632 0.5251 0.3368 0.2448 0.4660 0.3026 0.2219 ?= 0.5,?= 0.5,?= 5 0.8193 0.4787 0.3404 0.7079 0.4066 0.2885 0.6455 0.3788 0.2698 0.5771 0.3386 0.2420 ?= 0.9,?= 0.1,?= 5 1.1254 0.1355 0.0798 1.0412 0.1238 0.0729 0.9907 0.1195 0.0705 0.8700 0.1045 0.0615 In general, the lead time forecasts produced by an all-INAR(1) method beat the benchmark methods except for the cases when the data is sparse and the sample is small. This can be attributed to the fact that there is less positive data available for estimation of parameters for an INAR(1) process. The improvement increases with an increase in the length of history. When the lead time increases, the improvement of INARMA over benchmarks generally increases except for the case where the autoregressive parameter is high. BenchmarkINARMA MSEMSE / )( 6?l BenchmarkINARMA MSEMSE / )( 6?l M.Mohammadipour, 2009, Chapter 8 199 8.7 Conclusions In this chapter, the results of the simulation experiment have been presented. The performance of YW and CLS estimation methods (and CML for INAR(1) process) in terms of the accuracy of estimates and also their impact on forecast accuracy has been examined. The results show that when the length of history is short, CLS produces better forecasts than YW especially in the presence of a high autoregressive parameter. For cases where the autoregressive parameter is low, and also for the INMA(1) process, the two estimation methods are close. Also when the number of observations increases, the two estimation methods will produce close forecasts (in terms of MSE and MASE). The Croston-SBA categorization (Syntetos et al., 2005) has been tested and validated for an i.i.d. Poisson process (an INARMA(0,0) process). Although the categorization was originally developed using MSE of forecasts, the results of simulation show that it also holds when MASE of forecasts are considered. The simulation results show that the Croston-SBA categorization also holds for INAR(1), INMA(1) and INARMA(1,1) processes. It has been found that when the number of observation increases, the advantage of SBA over Croston in terms of MSE decreases until it reaches a limit. Four INARMA processes have been used in this study: INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1). The identification is therefore limited to selecting the best process among them. As discussed in chapter 4, two identification procedures are used. A two-stage identification procedure first uses the Ljung-Box statistic to distinguish between INARMA(0,0) and other processes. The AIC is then used to select among the other INARMA models. A one-stage identification procedure only uses AIC to select among all INARMA models including INARMA(0,0). The results show that the two-stage method provides better results for the INARMA(0,0) model (in terms of the percentage of series for which the correct model is identified). However, for other models, the one-stage method produces better results. In terms of the accuracy of forecasts, for an INARMA(0,0) process, the two-stage method produces better forecasts using MSE and MASE in most of the cases. For other processes, when the autoregressive parameter is high, the one-stage method produces M.Mohammadipour, 2009, Chapter 8 200 much better forecasts. This is also true for high moving average parameters but the difference is smaller. When more observations are available, the two methods produce similar forecasts. The results also suggest that, as expected, misidentification has a high effect on forecast accuracy when the autoregressive parameter is high. But for MA processes and for AR processes with low autoregressive parameters, the effect of misidentification is not high. As a potential substitute to identification, the most general INARMA model can be used. For example, if data is in fact an INAR(1) process, the estimated MA parameter should be close to zero. The results show that, in the presence of an AR component, using the most general INARMA model results in more accurate forecasts (in terms of MSE and MASE) than those of identification, especially for short history and high AR parameter. When the number of observations increases, the results of two methods will be close. We have also tested using an INAR(1) method to forecast all four INARMA models and have compared the results to the case of using an INARMA(1,1) method for all four models. The results show that the all-INAR(1) method generally produces better forecasts than the all-INARMA(1,1) method even for MA series. The INARMA forecasts are compared to those of Croston, SBA and SBJ. For INARMA(0,0) and INMA(1) processes, the improvement by using INARMA over benchmark methods is small. But when data is produced by INAR(1) or INARMA(1,1) and the autoregressive parameter is high, INARMA produces much more accurate one-step ahead forecasts than the benchmark methods. The degree of improvement generally increases when more observations are available. The results for three-step and six-step ahead forecasts show that, for INMA(1), and INAR(1) and INARMA(1,1) processes with small autoregressive parameters, the forecast accuracy of INARMA over benchmarks is improved compared to the one- step ahead forecasts. However, the same is not true for INARMA processes with high autoregressive parameters. Finally, the lead time forecasts of an all-INAR(1) method have been compared to those of the benchmark methods. The results show that the all-INAR(1) method generally beats the benchmark methods and the improvement increases when more M.Mohammadipour, 2009, Chapter 8 201 observations are available. The only exception is when both autoregressive parameter and lead time are high and the number of observations is small. In that case, the best benchmark method outperforms the all-INAR(1) method. Even for such cases, when the number of observations increases, INARMA starts to produce more accurate lead time forecasts than benchmarks. M.Mohammadipour, 2009, Chapter 9 202 Chapter 9 EMPIRICAL ANALYSIS 9.1 Introduction As discussed in chapter 3, INARMA models have been developed for forecasting count data. The application areas of these models have been mainly for counts of events or individuals such as the number of patients in a hospital?s emergency unit each hour. Intermittent series, as a series of non-negative integer values where some values are zero (Shenstone and Hyndman, 2005), can be considered as a special class of count series. However, there is no empirical evidence on the performance of INARMA models in this area. Accurate demand forecasting is a key to better inventory management. Although this M.Mohammadipour, 2009, Chapter 9 203 research only focuses on forecasting, an improvement in accuracy of forecasts can be translated to fewer inventories, less obsolescence and better customer service. This PhD thesis has suggested using INARMA models to forecast intermittent demand. The performance of these models has been compared to that of the benchmark methods of Croston, SBA and SBJ in Chapter 8. However, it was assumed that data were produced by one of the four INARMA models: INARMA(0,0), INAR(1), INMA(1) or INARMA(1,1). In this chapter, the model assumptions are relaxed by testing the results on empirical data. This chapter is organized as follows. The purposes of empirical analysis are explained in section 9.2. The series for this study consist of the demand data of 16,000 Royal Air Force (RAF) SKUs, some of which are highly lumpy, and 3,000 data series from the automotive industry. The filtering mechanism applied to the demand data for INARMA forecasting is discussed in section 9.3. Details of empirical analysis design are provided in section 9.4. The accuracy of INARMA forecasts and those of the benchmark methods are compared in section 9.5. In this section, the results of identification are compared with treating all as INAR(1) and INARMA(1,1). The ?-step ahead forecasts and lead time forecasts are also presented. The sensitivity of the results to the length of history has been tested. Finally, the conclusions of the empirical analysis are given in section 9.6. 9.2 Rationale for Empirical Analysis The main purpose of empirical analysis is to validate the theoretical and simulation findings on real data. The results of simulation show that when data is produced by INAR(1) or INARMA(1,1) and the autoregressive parameter is high, INARMA forecasting methods produce much more accurate forecasts than benchmark methods. But when data is produced by INARMA(0,0) or INMA(1), the improvement by using INARMA over benchmark methods is small. We are interested in finding out whether the INARMA forecasting approach still outperforms the benchmark methods for real intermittent demand data. In the simulation chapter we also looked at the effect of various factors on forecast M.Mohammadipour, 2009, Chapter 9 204 accuracy. The effect of YW and CLS (and CML for INAR(1)) estimates on the accuracy of INARMA forecasts was studied. The results of identification or using the most general INARMA model in the class were compared. The sample size effect on the accuracy of forecasts was also tested. The empirical analysis will assess the effect of these factors and enable us to validate the simulation results. 9.3 Demand Data Series The real demand data series for this research consists of the Royal Air Force (RAF) individual demand histories of 16,000 SKUs over a period of 6 years (monthly observations). We have also used another data set which consists of 3,000 real intermittent demand data series from the automotive industry1 (from Syntetos and Boylan, 2005) which, unlike the previous one, has more occurrences of positive demand than zeros. This data series consists of demand histories of 3,000 SKUs over a period of 2 years (24 months). These two data sets are called 16,000 and 3,000 series from now on. The 16,000 series are useful in assessing the effect of length of history on the accuracy of forecasts because it has longer history. However, it is can be categorized as a set of slower intermittent series because it has many periods of no demand. On the other hand, the 3,000 series has a very short history but it contains faster intermittent series with more positive demands. As previously mentioned, this research has focused on INARMA processes with Poisson innovations. Although some of the theoretical results are not based on a distributional assumption, whenever a specific distribution was needed, such as for estimation of parameters, a Poisson distribution was assumed. Out of the four processes of this study, three of them have a Poisson distribution when the marginal distribution is Poisson. The only exception is the INARMA(1,1) process where: ??=?????1 +??+?????1 Equation 9-1 1 This data set is available from: http://www.forecasters.org/ijf/data/Empirical%20Data.xls M.Mohammadipour, 2009, Chapter 9 205 ?????= 1 +? 1?? ? Equation 9-2 var????= 1 1??2 [1 +?+?+ 3??]? Equation 9-3 var???? ????? = 1 +?+?+ 3?? 1 +?+?+?? ?1.5 for 0???1, 0???1 Equation 9-4 In order to remove the data series with highly variable demands, a Poisson dispersion test (also called the variance test) is needed for all processes except INARMA(1,1). Under the null hypothesis that ?1,?,?? are Poisson distributioned, the test statistic: ???=? (????) 2 ? ? ?=1 Equation 9-5 has a chi-square distribution with (??1) degrees of freedom. Therefore, ?0 is rejected if ???>???1;1?? 2 . A revised statistic is used to allow for the difference between the mean and variance of an INARMA(1,1) process. The new test statistic is given by: ????= ??? 1.5 Equation 9-6 The new statistic also has a chi-square distribution with (??1) degrees of freedom. Further filtering of data was performed for series with fewer than two nonzero demands. As previously mentioned in Chapter 8, the benchmark methods need at least two nonzero observations for initialization. Out of the 16,000 series, 5,168 series met the above criteria and therefore are used for empirical analysis. The filtering of the 3,000 series results in 1,943 series. It can be seen that although a substantial number of series has the potential to benefit from PoINARMA models, for a large number of series these models are not appropriate. M.Mohammadipour, 2009, Chapter 9 206 Other distributional assumptions would obviously result in different number of filtered series, which can be pursued as a further study. 9.4 Design of Empirical Analysis The design of the empirical analysis follows the detailed simulation design of Chapter 7. As discussed in section 7.3.3.4, two fixed values have been used for the smoothing parameter of Croston, SBA, and SBJ methods (?= 0.2, 0.5). The initialization for these methods is based on using the first inter-demand interval as the first smoothed inter-demand interval and the average of the first two non-zero observations as the first smoothed size. The data series is divided into two parts: ?estimation period? for initialization and estimation of parameters and ?performance period? for assessing the accuracy of forecasts. If at least two non-zero demands are observed in the estimation period, the first half of the observations is assigned for the estimation period and the other half for the performance period. However, if fewer than two non-zero demands are observed in the estimation period, this period will be extended until the second non- zero demand is observed. When the effect of length of history is tested, the performance period is fixed and the estimation period varies. It is assumed that there are four possible INARMA models to use for forecasting. Therefore, identification is undertaken among these models. This is done by applying both two-stage and one-stage identification procedures (see section 4.6). The former first uses the Ljung-Box test to distinguish between INARMA(0,0) and the other three models. Then, the AIC is used to select among the other models. The latter uses only the AIC to select among all INARMA models. Based on the simulation results of chapter 8, we suggested that general models (INAR(1) and INARMA(1,1) were tested in chapter 8) can be used as alternatives to identification. This is also tested on empirical data. YW and CLS (and CML only for the INAR(1) model) have been used to estimate the parameters of INARMA models. The simulation results show that these methods M.Mohammadipour, 2009, Chapter 9 207 result in similar forecasts when the length of history is high. But for short history and high autoregressive parameters, CLS generally produces more accurate one-step ahead forecasts in terms of MSE and MASE. These estimation methods have been used for empirical analysis to test the validity of the simulation findings. It is worth mentioning that the INARMA forecasts are minimum mean square (MMSE). Therefore, we expect an improvement in terms of MSE but not necessarily in terms of MASE. The accuracy measures used in empirical analysis are discussed in chapter 2 in detail. In addition to the measures used in the simulation experiment (ME, MSE, and MASE), two relative-to-another-method measures have been used. As previously mentioned in section 2.4, the percent better (PB) determines how often a method is better than another method. However, it does not show how much the improvement is. The relative geometric root-mean-square error (RGRMSE) is used to calculate the magnitude of improvement of one method over another. 9.5 INARMA vs Benchmark Methods In this section, the results of comparing INARMA forecasts with those of benchmarks are presented. First, we use an all-INAR(1) approach assuming that all the series are in fact INAR(1) processes. The different estimation methods used in the simulation chapter for estimation of INAR(1) parameters are also used here. An all-INARMA(1,1) approach is then used and the results are compared to both benchmark methods and all-INAR(1) results. As previously mentioned in sections 8.6.2.2 and 8.6.2.4, these approaches especially perform well compared to identification for highly auto-correlated data and short data histories. To assess their performance for empirical data, the forecast accuracy results based on identification are also presented. The results of identification show what INARMA models each series follows. Those series that follow a specific INARMA model are then separated and each one is forecasted with the corresponding INARMA method (either INARMA(0,0), INAR(1), INMA(1), or INARMA(1,1)). The accuracy of these INARMA forecasts is compared to the corresponding simulation results. The results include one-step, M.Mohammadipour, 2009, Chapter 9 208 three-step and six-step-ahead forecasts. The sensitivity of INARMA forecasts to the length of history is also tested for the 16,000 series. Finally, the results of lead time forecasts for the INAR(1) model are presented for both the 16,000 and 3,000 series. 9.5.1 All-INAR(1) We suggest in chapter 8 that using a general INARMA model produces forecasts comparable to those based on identification. As we will show in section 9.5.3, most of the series in both 16,000 and 3,000 series are identified as INARMA(0,0) or INAR(1). Therefore, using INAR(1) to forecast seems to be a promising approach for these datasets. The results of INAR(1) and benchmarks for all points in time and issue points are shown in Table 9-1 and Table 9-2 for 16,000 series. Recall from chapter 5 that, as pointed out by Al-Osh and Alzaid (1987), for small sample sizes (??75) and small autoregressive parameter ??= 0.1?, because the sample contains many zero values, CML is not as good as YW in terms of bias and MSE of the estimates. The simulation results in section 8.3 show that for corresponding parameters (see section 9.5.4), YW yields better forecasts than CLS and CML using MSE and MASE. We test this by presenting CLS, YW and CML based forecasts. The empirical results confirm the corresponding simulation results. Table ?9-1 Comparing INAR(1) with benchmarks for all points in time (16000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INAR(1) CLS INAR(1) YW INAR(1) CML ME -0.0719 -0.0933 -0.0402 -0.0086 -0.0367 0.0196 0.0261 0.0283 0.0282 MSE 0.3910 0.4205 0.3802 0.3859 0.3793 0.3846 0.3609 0.3527 0.3555 MASE 2.8594 2.8852 2.7051 2.4925 2.6881 2.3640 1.9789 1.9124 1.9432 PB of MASE INAR(1)-CLS/Benchmark 0.6830 0.6952 0.6302 0.5624 0.6236 0.5116 PB of MASE INAR(1)-YW/Benchmark 0.6742 0.6868 0.6216 0.5555 0.6150 0.5050 PB of MASE INAR(1)-CML/Benchmark 0.6651 0.6878 0.6116 0.5545 0.6050 0.5050 RGRMSE INAR(1)-CLS/Benchmark 0.7812 0.7401 0.8272 0.9024 0.8352 0.9927 RGRMSE INAR(1)-YW/Benchmark 0.7898 0.7493 0.8382 0.9157 0.8463 1.0077 RGRMSE INAR(1)-CML/Benchmark 0.7894 0.7494 0.8367 0.9152 0.8449 1.0075 M.Mohammadipour, 2009, Chapter 9 209 Table ?9-2 Comparing INAR(1) with benchmarks for issue points (16000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INAR(1) CLS INAR(1) YW INAR(1) CML ME -0.0266 -0.0441 0.0028 0.0339 0.0061 0.0598 -0.0598 -0.0324 -0.0197 MSE 0.5329 0.5666 0.5237 0.5344 0.5231 0.5359 0.5792 0.5218 0.5420 MASE 0.3309 0.3375 0.3188 0.3048 0.3175 0.2951 0.3679 0.3473 0.3444 PB of MASE INAR(1)-CLS/Benchmark 0.4590 0.4509 0.4016 0.3219 0.3949 0.2900 PB of MASE INAR(1)-YW/Benchmark 0.4584 0.4493 0.4026 0.3273 0.3956 0.2941 PB of MASE INAR(1)-CML/Benchmark 0.4826 0.4785 0.4234 0.3534 0.4166 0.3196 RGRMSE INAR(1)-CLS/Benchmark 1.2416 1.3295 1.3402 1.6694 1.3531 1.8393 RGRMSE INAR(1)-YW/Benchmark 1.1734 1.2477 1.2728 1.5709 1.2847 1.7301 RGRMSE INAR(1)-CML/Benchmark 1.1060 1.2158 1.1989 1.5367 1.2105 1.6960 The results of Table 9-1 show that INAR(1) produces better results than benchmarks regardless of the estimation method. The results also show that YW based INAR(1) is the best estimation method among the three methods, outperforming the best benchmark method (SBJ 0.2 for MSE and SBJ 0.5 for MASE) by 7 percent in terms of MSE and 19 percent in terms of MASE. The results of PB and RGRMSE also show superior performance of INARMA compared to the benchmark methods. As can be seen from Table 9-2, when only issue points are considered, the INARMA forecasts are biased, agreeing with the simulation results. We find that this is true for all the cases that only issue points are considered so we do not discuss this again. It can be seen from Table 9-1 that the MASE of all methods is very high which, as explained in section 2.4, suggests that all of these methods are worse than na?ve. The reason is that because the data series contain many zeros in the estimation period, the error of na?ve in most of the periods is zero. Therefore the in-sample MAE is very small and the MASE is very large. However, when only issue points are considered, because it is likely that a nonzero demand is followed with a zero demand, the absolute error of na?ve and therefore the in-sample MAE is large. As a result, the MASE of the forecasting methods is smaller compared to the all points in time case. The results of Table 9-2 confirm this. As a result, for highly intermittent data, MASE does not provide reliable results for M.Mohammadipour, 2009, Chapter 9 210 all points in time and issue points. The simulation results of section 8.6.1 show the superiority of INARMA(0,0) forecasts compared to the benchmarks for ?= 0.3,?= 96. It will be seen in section 9.5.4 that the majority of 16,000 series are identified as INARMA(0,0) with ? close to 0.3 (see appendix 9.A) so the results for 16,000 series for all points in time agree with the simulation results. The results for 3,000 series are shown in Table 9-3 and Table 9-4. It can be seen that YW again results in more accurate forecasts than both CLS and CML. CLS and CML results are worse than benchmarks and YW results only improve MSE by 0.5 percent compared to the best benchmark which is SBA 0.2. However, the MASE of SBJ 0.2 is better than that of INAR(1)-YW by one percent. As mentioned in chapter 6, the INARMA forecasts provide the minimum MSE and not MASE. The results of Table 9-3 also show the superior performance of INAR(1)-YW to the benchmark methods in terms of PB and RGRMSE except for SBA and SBJ 0.2. Based on the results of Table 9-4, INAR(1)-YW is better than the best benchmark in terms of MSE although the INARMA forecasts are biased when only issue points are considered. However, the MASE is still slightly worse. Table ?9-3 Comparing INAR(1) with benchmarks for all points in time (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INAR(1) CLS INAR(1) YW INAR(1) CML ME -0.0419 -0.1016 0.1662 0.4336 0.1894 0.6120 0.1016 0.0136 0.0145 MSE 3.2574 3.7054 3.2483 3.6470 3.2550 3.8249 3.3540 3.2319 3.2640 MASE 0.8694 0.9277 0.8543 0.8848 0.8535 0.8914 0.8757 0.8636 0.8720 PB of MASE INAR(1)-CLS/Benchmark 0.5097 0.5528 0.4885 0.5211 0.4886 0.5329 PB of MASE INAR(1)-YW/Benchmark 0.5171 0.5551 0.4842 0.5211 0.4839 0.5295 PB of MASE INAR(1)-CML/Benchmark 0.5133 0.5522 0.4828 0.5136 0.4812 0.5259 RGRMSE INAR(1)-CLS/Benchmark 0.5329 0.9814 0.9370 1.0027 1.0037 1.0067 RGRMSE INAR(1)-YW/Benchmark 0.9471 0.9175 0.9765 0.9868 0.9826 0.9857 RGRMSE INAR(1)-CML/Benchmark 0.9476 0.9207 0.9791 0.9898 0.9847 0.9894 M.Mohammadipour, 2009, Chapter 9 211 Table ?9-4 Comparing INAR(1) with benchmarks for issue points (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INAR(1) CLS INAR(1) YW INAR(1) CML ME -0.0518 -0.1058 0.1544 0.4233 0.1773 0.5996 -0.0037 -0.0650 -0.0732 MSE 3.3807 3.8368 3.3685 3.7724 3.3749 3.9483 3.4687 3.3563 3.3865 MASE 0.9083 0.9686 0.8918 0.9210 0.8907 0.9260 0.9164 0.9054 0.9130 PB of MASE INAR(1)-CLS/Benchmark 0.5087 0.5555 0.4861 0.5157 0.4857 0.5231 PB of MASE INAR(1)-YW/Benchmark 0.5117 0.5539 0.4756 0.5127 0.4744 0.5161 PB of MASE INAR(1)-CML/Benchmark 0.5097 0.5519 0.4741 0.5076 0.4723 0.5162 RGRMSE INAR(1)-CLS/Benchmark 1.0187 0.9738 1.0354 1.0442 1.0434 1.0503 RGRMSE INAR(1)-YW/Benchmark 0.9927 0.9618 1.0200 1.0311 1.0293 1.0377 RGRMSE INAR(1)-CML/Benchmark 0.9953 0.9686 1.0291 1.0434 1.0394 1.0495 The simulation results of chapter 8 suggest that, with the presence of a high autocorrelation, INAR(1) outperforms the benchmarks. Also, the simulation results show that for ?= 0.1 and ?= 1, 3, when the number of observations is small (?= 24), SBA and SBJ (with smoothing parameter 0.2) are better than INAR(1) in terms of both MSE and MASE. But for higher number of observations, INAR(1) starts to slightly perform better. It will be seen in section 9.5.4 that the estimated autoregressive parameter is generally close to 0.1. Therefore, the results for 3000 series agree with the simulation results. 9.5.2 All-INARMA(1,1) In this section it is assumed that all series follow the INARMA(1,1) model. The results for 16,000 series are presented in Table 9-5 and Table 9-6. The simulation results show that for INARMA(1,1) processes, CLS produces better results than YW. We test this for empirical data by presenting both CLS and YW- based forecasts. As explained in chapter 5, the CML estimates have only been obtained for INAR(p) models and therefore these estimates are not presented for the INARMA(1,1) model. M.Mohammadipour, 2009, Chapter 9 212 The results of Table 9-5 show that INARMA improves the accuracy of forecast by 3 percent in terms of MSE and 19 percent in terms of MASE compared to the best benchmark (SBJ 0.2 for MSE and SBJ 0.5 for MASE). The RGRMSE results confirm the MSE results. It can be seen that the YW-based INARMA(1,1) forecasts are worse than CLS-based forecasts, agreeing with simulation results. Table ?9-5 Comparing INARMA(1,1) with benchmarks for all points in time (16000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INARMA CLS INARMA YW ME -0.0719 -0.0933 -0.0402 -0.0086 -0.0367 0.0196 0.0258 -0.0039 MSE 0.3910 0.4205 0.3802 0.3859 0.3793 0.3846 0.3668 0.4833 MASE 2.8594 2.8852 2.7051 2.4925 2.6881 2.3640 1.9128 2.3085 PB of MASE INARMA-CLS/Benchmark 0.6692 0.6842 0.6161 0.5508 0.6095 0.5004 PB of MASE INARMA-YW/Benchmark 0.7359 0.7413 0.7150 0.6922 0.7122 0.6691 RGRMSE INARMA-CLS/Benchmark 0.7653 0.7294 0.8128 0.8918 0.8205 0.9818 RGRMSE INARMA-YW/Benchmark 0.7843 0.7464 0.8200 0.8960 0.8285 0.9881 Table ?9-6 Comparing INARMA(1,1) with benchmarks for issue points (16000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INARMA CLS INARMA YW ME -0.0266 -0.0441 0.0028 0.0339 0.0061 0.0598 -0.0326 -0.5620 MSE 0.5329 0.5666 0.5237 0.5344 0.5231 0.5359 0.5766 1.5057 MASE 0.3309 0.3375 0.3188 0.3048 0.3175 0.2951 0.3509 0.7313 PB of MASE INARMA-CLS/Benchmark 0.4958 0.4797 0.4349 0.3507 0.4275 0.3162 PB of MASE INARMA-YW/Benchmark 0.3004 0.2923 0.2779 0.2375 0.2750 0.2185 RGRMSE INARMA-CLS/Benchmark 1.0833 1.1712 1.1751 1.4768 1.1862 1.6282 RGRMSE INARMA-YW/Benchmark 2.1539 2.3492 2.4021 2.9709 2.4289 3.3006 The results of comparing INARMA(1,1) and benchmarks for all points in time and issue points for 3,000 series are presented in Table 9-7 and Table 9-8 for both CLS and YW estimates. The results confirm that INARMA(1,1) forecasts based on CLS estimates are better than those based on YW estimates. The results of Table 9-7 show that both CLS and YW based INARMA(1,1) forecasts are worse than benchmarks. In order to understand the reason, we need to identify M.Mohammadipour, 2009, Chapter 9 213 the autoregressive and moving average order of the data series. This will be examined in the next section. Table ?9-7 Comparing INARMA(1,1) with benchmarks for all points in time (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INARMA CLS INARMA YW ME -0.0419 -0.1016 0.1662 0.4336 0.1894 0.6120 -0.0461 0.0091 MSE 3.2574 3.7054 3.2483 3.6470 3.2550 3.8249 3.4038 4.0726 MASE 0.8694 0.9277 0.8543 0.8848 0.8535 0.8914 0.8869 0.9548 PB of MASE INARMA-CLS/Benchmark 0.5063 0.5401 0.4777 0.5125 0.4767 0.5209 PB of MASE INARMA-YW/Benchmark 0.4734 0.5018 0.4517 0.4782 0.4522 0.4899 RGRMSE INARMA-CLS/Benchmark 0.9893 0.9536 1.0231 1.0293 1.0289 1.0276 RGRMSE INARMA-YW/Benchmark 0.7435 0.7078 0.7675 0.7680 0.7723 0.7669 Table ?9-8 Comparing INARMA(1,1) with benchmarks for issue points (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INARMA CLS INARMA YW ME -0.0518 -0.1058 0.1544 0.4233 0.1773 0.5996 -0.1366 -0.1972 MSE 3.3807 3.8368 3.3685 3.7724 3.3749 3.9483 3.5569 4.1304 MASE 0.9083 0.9686 0.8918 0.9210 0.8907 0.9260 0.9327 0.9902 PB of MASE INARMA-CLS/Benchmark 0.5006 0.5378 0.4691 0.5027 0.4677 0.5097 PB of MASE INARMA-YW/Benchmark 0.4805 0.5108 0.4558 0.4844 0.4570 0.4961 RGRMSE INARMA-CLS/Benchmark 1.0343 0.9940 1.0629 1.0707 1.0712 1.0772 RGRMSE INARMA-YW/Benchmark 0.8166 0.7749 0.8417 0.8469 0.8486 0.8527 The simulation results for the corresponding parameter set (?= 0.1, 0.3, ? close to zero, and ?= 1,3) show that when the number of observations is small (?= 24), INARMA(1,1) performance is poor. But for a higher autoregressive parameter and number of observations, INARMA(1,1) performance improves greatly. Comparing the results of this section with those of the previous section shows that treating all as INAR(1) produces better results than treating all as INARMA(1,1) which confirms the simulation results (see section 8.6.2.4). A possible explanation is that the number of parameters to be estimated and therefore the estimation error is less for all-INAR(1) compared to all-INARMA(1,1). The results of identification in M.Mohammadipour, 2009, Chapter 9 214 the next section will show that 98.78 percent of 16,000 series (78.43 for 3,000 series) are identified as INAR(0,0) or INAR(1). This could also justify the superior performance of all-INAR(1) because only 1.22 percent of the series (21.57 for 3000 series) are either INMA(1) or INARMA(1,1). Although more series are identified as INMA(1) and INARMA(1,1) in 3,000 series, their moving average parameter is generally between zero and 0.1 and these processes are close to INARMA(0,0) and INAR(1). This could also help to understand why all-INAR(1) works better than all- INARMA(1,1) for 3,000 series. 9.5.3 Identification among four Processes In this section, the appropriate INARMA model is identified among the four possible candidates. Both one-stage and two-stage identification procedures are tested (see section 4.6 for details). For both identification methods, the results for the case where all the INARMA forecasts are based on CLS and YW estimates are presented. The accuracy of INARMA forecasts based on identification and treating all as INAR(1) or INARMA(1,1) for 16,000 series are compared in Table 9-9 and Table 9-10. As expected, identification (two-stage) produces slightly better results in terms of MSE and MASE, (except for MSE of all-INAR(1) for all points in time) but the results are generally close. Table ?9-9 The effect of identification on INARMA forecasts for all points in time (16000 series) The identification method ME MSE MASE PB of MASE (INARMA/Benchmark) RGRMSE (INARMA/Benchmark) Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Two-stage identification (CLS) 0.0334 0.3529 1.8576 0.6659 0.6818 0.6099 0.5440 0.6030 0.4927 0.8004 0.7614 0.8514 0.9328 0.8595 1.0264 Two-stage identification (YW) 0.0321 0.3521 1.8647 0.6655 0.6811 0.6096 0.5440 0.6027 0.4924 0.7968 0.7577 0.8474 0.9286 0.8555 1.0217 One-stage identification (CLS) 0.0307 0.3581 1.9008 0.6665 0.6832 0.6123 0.5481 0.6054 0.4963 0.7948 0.7549 0.8443 0.9232 0.8524 1.0165 One-stage identification (YW) 0.0309 0.3533 1.9085 0.6359 0.6413 0.6150 0.5422 0.6022 0.4961 0.7945 0.7557 0.8436 0.9221 0.8529 1.0151 All-INAR(1) (YW) 0.0283 0.3527 1.9124 0.6742 0.6868 0.6216 0.5555 0.6150 0.5050 0.7898 0.7493 0.8382 0.9157 0.8463 1.0077 All-INARMA(1,1) (CLS) 0.0258 0.3668 1.9128 0.6692 0.6842 0.6161 0.5508 0.6095 0.5004 0.7653 0.7294 0.8128 0.8918 0.8205 0.9818 M.Mohammadipour, 2009, Chapter 9 215 Table ?9-10 The effect of identification on INARMA forecasts for issue points (16000 series) The identification method ME MSE MASE PB of MASE (INARMA/Benchmark) RGRMSE (INARMA/Benchmark) Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Two-stage identification (CLS) 0.0351 0.5045 0.3055 0.5562 0.5246 0.4903 0.3836 0.4817 0.3448 0.9315 0.9659 1.0093 1.2102 1.0186 1.3311 Two-stage identification (YW) 0.0256 0.5187 0.3093 0.5514 0.5196 0.4856 0.3795 0.4768 0.3406 0.9587 0.9910 1.0392 1.2435 1.0488 1.3688 One-stage identification (CLS) 0.0012 0.5288 0.3293 0.5211 0.5014 0.4581 0.3682 0.4497 0.3308 1.0188 1.0941 1.1053 1.3783 1.1158 1.5188 One-stage identification (YW) 0.0220 0.5257 0.3313 0.5204 0.5023 04579 0.3575 0.4450 0.3185 1.0145 1.0722 1.0951 1.3723 1.1116 1.5252 All-INAR(1) (YW) -0.0324 0.5218 0.3473 0.4584 0.4493 0.4026 0.3273 0.3956 0.2941 1.1734 1.2477 1.2728 1.5709 1.2847 1.7301 All-INARMA(1,1) (CLS) -0.0326 0.5766 0.3509 0.4958 0.4797 0.4349 0.3507 0.4275 0.3162 1.0833 1.1712 1.1751 1.4768 1.1862 1.6282 The results of Table 9-9 and Table 9-10 also show that the two-stage identification procedure provides better results than the one-stage. Since the majority of series are identified as INARMA(0,0), this agrees with the simulation results of section 8.6.2.3. It can also be seen that the CLS and YW yield close results for both two-stage and one-stage identification methods. In order to be consistent with the simulation analysis (see conclusion of section 8.4), we focus on the CLS-based two-stage identification results. For 16,000 time series, out of the 5,168 series, 98.12 percent were identified as INARMA(0,0), 0.66 percent as INAR(1), 1.04 percent as INMA(1), and 0.17 percent were identified as INARMA(1,1). As can be seen, the majority of the series are identified as INARMA(0,0). The simulation results show that when data is in fact INARMA(0,0), the all- INARMA(1,1) forecasts are close to INARMA(0,0) forecasts. The simulation results also show that when the order is known to be (0,0), all-INARMA(1,1) produces better forecasts than the best benchmark method. The results of Table 9-5 and Table 9-6 show that all-INARMA(1,1) for 16,000 series performs better than benchmarks, agreeing with the simulation results. Now, the accuracy of INARMA forecasts based on identification and treating all as INAR(1) or INARMA(1,1) for 3,000 series are compared in Table 9-11 and Table 9-12. Here again, the two-stage identification produces slightly better results not only M.Mohammadipour, 2009, Chapter 9 216 in terms of MSE and MASE, but also in terms of PB of MASE and RGRMSE (with the exception of MSE of all-INAR(1)). It should be mentioned that these results are still not as good as SBA and SBJ with smoothing parameter 0.2. Table ?9-11 The effect of identification on INARMA forecasts for all points in time (3000 series) The identification method ME MSE MASE PB of MASE (INARMA/Benchmark) RGRMSE (INARMA/Benchmark) Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Two-stage identification (CLS) 0.0813 3.2925 0.8608 0.5238 0.5557 0.4973 0.5286 0.4974 0.5358 0.9376 0.9050 0.9599 0.9710 0.9647 0.9707 Two-stage identification (YW) 0.0191 3.2840 0.8603 0.5229 0.5514 0.4895 0.5244 0.4893 0.5315 0.9135 0.8875 0.9412 0.9544 0.9475 0.9500 One-stage identification (CLS) 0.0276 3.3088 0.8697 0.5156 0.5497 0.4852 0.5191 0.4854 0.5276 0.9469 0.9145 0.9747 0.9829 0.9803 0.9823 One-stage identification (YW) 0.0091 3.3726 0.8748 0.4734 0.5018 0.4517 0.4782 0.4522 0.4899 0.9475 0.9178 0.9679 0.9980 0.9823 0.9969 All-INAR(1) (YW) 0.0136 3.2319 0.8636 0.5171 0.5551 0.4842 0.5211 0.4839 0.5295 0.9471 0.9175 0.9765 0.9868 0.9826 0.9857 All-INARMA(1,1) (CLS) -0.0461 3.4038 0.8869 0.5063 0.5401 0.4777 0.5125 0.4767 0.5209 0.9893 0.9536 1.0231 1.0293 1.0289 1.0276 Table ?9-12 The effect of identification on INARMA forecasts for issue points (3000 series) The identification method ME MSE MASE PB of MASE (INARMA/Benchmark) RGRMSE (INARMA/Benchmark) Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Crost 0.2 Crost 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Two-stage identification (CLS) 0.0099 3.4099 0.9013 0.5202 0.5544 0.4888 0.5225 0.4883 0.5256 0.9907 0.9560 1.0076 1.0214 1.0160 1.0277 Two-stage identification (YW) -0.0303 3.3835 0.8995 0.5190 0.5501 0.4827 0.5191 0.4814 0.5225 0.9653 0.9365 0.9879 1.0004 0.9973 1.0039 One-stage identification (CLS) -0.0490 3.4327 0.9107 0.5107 0.5486 0.4783 0.5127 0.4780 0.5174 0.9961 0.9618 1.0188 1.0301 1.0282 1.0371 One-stage identification (YW) -0.0572 3.5304 0.9202 0.5005 0.5408 0.4658 0.5144 0.4670 0.5161 0.9966 0.9749 1.0217 1.0469 1.0486 1.0527 All-INAR(1) (YW) -0.0650 3.3563 0.9054 0.5117 0.5539 0.4756 0.5127 0.4744 0.5161 0.9927 0.9618 1.0200 1.0311 1.0293 1.0377 All-INARMA(1,1) (CLS) -0.1366 3.5569 0.9327 0.5006 0.5378 0.4691 0.5027 0.4677 0.5097 1.0343 0.9940 1.0629 1.0707 1.0712 1.0772 The results of Table 9-11 and Table 9-12 also show that the two-stage identification procedure provides better results than the one-stage method. This agrees with the corresponding simulation results of section 8.6.2.3 for each of INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) models with similar parameters to those estimated for 3,000 series (see section 9.5.4 to find the corresponding parameters). The CLS-based two-stage identification results for 3,000 series show that out of 1,943 filtered series, 54.55 percent were identified as INARMA(0,0), 23.88 percent M.Mohammadipour, 2009, Chapter 9 217 as INAR(1), 17.96 percent as INMA(1), and 3.60 percent were identified as INARMA(1,1). Now, if we go back to the results of the previous section, the results show that all- INARMA(1,1) results are worse than the benchmark methods. The simulation results also show that for the INAR(1) and INMA(1) processes with the corresponding parameters (the most common INARMA parameters of the 1,943 series are provided in the next section), the INARMA forecast accuracy is very close or slightly worse than that of benchmark methods. This could explain the 4 percent superiority (in terms of MSE) of the best benchmark forecasts (SBA 0.2) over the all- INARMA(1,1) forecasts. Based on the results of this section, identification (two-stage) results in better INARMA forecasts than using an all-INAR(1) or all-INARMA(1,1) approach but the accuracy benefits are small. As also mentioned in the simulation chapter, the all- INAR(1) and all-INARMA(1,1) approaches are especially useful when the autoregressive parameter is high, but this is not the case for our empirical data. 9.5.4 INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) Series As previously mentioned, a considerable percentage of series among 1,943 filtered series (of 3,000 series) were identified as INARMA(0,0), INAR(1) or INMA(1) and a few series were identified as INARMA(1,1). In this section, we separate the series according to the models identified and study them individually. Because most of the series in the 16,000 data set are identified as INARMA(0,0) and a small number identified as other models, we do not present the results of each model here. Similar results for 16,000 series can be found in Appendix 9.A. As the results of identification in the previous section suggest, 54.55 percent of 1943 series are identified as INARMA(0,0). Now, we assume that the order of these 1,060 series is taken to be (0,0) and the INARMA(0,0) forecasting method, i.e. using the average of all the previous observations as the forecast for the next period, is used. The results for all points in time and issue points are presented in Table 9-13 and Table 9-14. M.Mohammadipour, 2009, Chapter 9 218 Table ?9-13 Only INARMA(0,0) series for all points in time (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INARMA ME 0.0288 -0.0843 0.1719 0.3017 0.1878 0.4303 0.1420 MSE 2.0878 2.3629 2.0998 2.2830 2.1043 2.3483 2.0749 MASE 0.9139 0.9819 0.9021 0.9352 0.9016 0.9382 0.8978 PB of MASE (INARMA/Benchmark) 0.5329 0.5624 0.5080 0.5453 0.5085 0.5491 - RGRMSE (INARMA/Benchmark) 0.8992 0.8543 0.9111 0.8922 0.9152 0.8798 - Table ?9-14 Only INARMA(0,0) series for issue points (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INARMA ME 0.0071 -0.0959 0.1482 0.2827 0.1639 0.4089 0.0830 MSE 2.1982 2.4911 2.2013 2.3901 2.2049 2.4482 2.1704 MASE 0.9472 1.0182 0.9327 0.9635 0.9318 0.9634 0.9298 PB of MASE (INARMA/Benchmark) 0.5328 0.5642 0.5023 0.5388 0.5013 0.5362 - RGRMSE (INARMA/Benchmark) 0.9743 0.9149 0.9763 0.9549 0.9836 0.9450 - Investigating the estimated parameter of the INARMA(0,0) process (?), we found that in general ? is close to 1 (the average is 1.2641 and 73.87 percent are between 0.5 and 1.5). The results of Table 9-13 agree with the corresponding simulation results (?= 1 and ?= 24). The simulation results show that INARMA produces the best results and the empirical results also show a very narrow improvement over the best benchmark method. The PB of MASE results confirm the results of MASE for both all points in time and issue points. The results of identification show that 23.88 percent of 1,943 series are identified as INAR(1). Now, we assume that the order of these 464 series is taken to be (1,0) and the INAR(1) forecasting method is used. We also investigate the effect of using different estimation methods on forecast accuracy. For this reason, the forecasts based on the three estimation methods of CLS, YW and CML are presented in Table 9-15 and Table 9-16. M.Mohammadipour, 2009, Chapter 9 219 In general, the estimated autoregressive parameter of the INAR(1) process, ?, is close to 0.1 (the average is 0.1234 and 50.65 percent are between 0.05 and 0.15) and the estimated innovation parameter, ?, is around 2 (the average is 2.5972 and 40.52 percent are between 1 and 3). The simulation results for the corresponding parameters (similar values of ?, ?, and ?) show that YW yields better results than CLS and CML. The simulation results also show that the YW results are slightly better than those of SBA and SBJ with smoothing parameter 0.2. Table ?9-15 Only INAR(1) series for all points in time (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INAR(1) CLS INAR(1) YW INAR(1) CML ME -0.1557 -0.1498 0.1365 0.5791 0.1689 0.8221 -0.1963 -0.1411 -0.1257 MSE 4.8743 5.6076 4.8235 5.5217 4.8319 5.8396 4.9049 4.7921 4.8386 MASE 0.9168 0.9721 0.8923 0.9218 0.8906 0.9312 0.9203 0.9088 0.9177 PB of MASE INAR(1)-CLS/Benchmark 0.4989 0.5408 0.4596 0.4919 0.4605 0.5068 PB of MASE INAR(1)-YW/Benchmark 0.5119 0.5458 0.4682 0.5038 0.4662 0.5135 PB of MASE INAR(1)-CML/Benchmark 0.5083 0.5462 0.4657 0.4928 0.4653 0.5070 RGRMSE INAR(1)-CLS/Benchmark 0.9940 0.9733 1.0499 1.0924 1.0529 1.1180 RGRMSE INAR(1)-YW/Benchmark 0.9726 0.9554 1.0275 1.0703 1.0311 1.0950 RGRMSE INAR(1)-CML/Benchmark 0.9887 0.9672 1.0443 1.0835 1.0457 1.1087 Table ?9-16 Only INAR(1) series for issue points (3000 series) Accuracy measure Forecasting method Croston 0.2 Croston 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INAR(1) CLS INAR(1) YW INAR(1) CML ME -0.1439 -0.1390 0.1465 0.5856 0.1787 0.8272 -0.2755 -0.2165 -0.2114 MSE 5.0350 5.7658 4.9933 5.7037 5.0027 6.0297 5.1167 4.9863 5.0296 MASE 0.9600 1.0166 0.9362 0.9669 0.9346 0.9764 0.9743 0.9609 0.9685 PB of MASE INAR(1)-CLS/Benchmark 0.4850 0.5330 0.4464 0.4837 0.4473 0.4962 PB of MASE INAR(1)-YW/Benchmark 0.4988 0.5411 0.4579 0.4954 0.4551 0.5024 PB of MASE INAR(1)-CML/Benchmark 0.5005 0.5413 0.4544 0.4872 0.4552 0.5001 RGRMSE INAR(1)-CLS/Benchmark 1.0272 1.0087 1.0864 1.1242 1.0913 1.1743 RGRMSE INAR(1)-YW/Benchmark 1.0041 0.9888 1.0625 1.1012 1.0682 1.1496 RGRMSE INAR(1)-CML/Benchmark 1.0177 0.9981 1.0731 1.1107 1.0778 1.1582 The results of Table 9-15 agree with the simulation results in that YW results are M.Mohammadipour, 2009, Chapter 9 220 better than CLS and CML. The YW-based INAR(1) forecasts are also better than benchmarks by one percent in terms of MSE. However, the MASE of INAR(1) is worse than that of the best benchmark by one percent. The results of Table 9-16 also show a narrow improvement of INAR(1)-YW compared to the best benchmark (0.15 percent in terms of MSE) although the INARMA forecasts are biased when only issue points are considered. Based on the results of identification, 17.96 percent of 1,943 series are identified as INMA(1). We assume that the order of these 349 series is taken to be (0,1) and the INMA(1) forecasting method is used. We also investigate the effect of using different estimation methods (CLS and YW) on the forecasting accuracy. The results are given in Table 9-17 and Table 9-18. Table ?9-17 Only INMA(1) series for all points in time (3000 series) Accuracy measure Forecasting method Croston 0.2 Crosto n 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INMA(1) CLS INMA(1) YW ME -0.0869 -0.0875 0.1907 0.6066 0.2215 0.8380 -0.1041 -0.0936 MSE 4.3556 4.9339 4.3470 4.9673 4.3581 5.2745 4.3291 4.3464 MASE 0.6681 0.7022 0.6556 0.6778 0.6549 0.6904 0.6676 0.6668 PB of MASE INMA(1)-CLS/Benchmark 0.5093 0.5418 0.4728 0.5033 0.4702 0.5150 PB of MASE INMA(1)-YW/Benchmark 0.5150 0.5461 0.4742 0.5062 0.4723 0.5177 RGRMSE INMA(1)-CLS/Benchmark 0.9546 0.9687 0.9829 1.0729 0.9960 1.0516 RGRMSE INMA(1)-YW/Benchmark 0.9345 0.9527 0.9642 1.0524 0.9771 1.0316 Table ?9-18 Only INMA(1) series for issue points (3000 series) Accuracy measure Forecasting method Croston 0.2 Crosto n 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INMA(1) CLS INMA(1) YW ME -0.0870 -0.0846 0.1889 0.6043 0.2195 0.8340 -0.1593 -0.1444 MSE 4.4429 5.0046 4.4358 5.0503 4.4471 5.3611 4.4548 4.4742 MASE 0.7177 0.7512 0.7045 0.7255 0.7039 0.7391 0.7256 0.7235 PB of MASE INMA(1)-CLS/Benchmark 0.4924 0.5304 0.4575 0.4891 0.4539 0.5019 PB of MASE INMA(1)-YW/Benchmark 0.5026 0.5361 0.4616 0.4945 0.4594 0.5064 RGRMSE INMA(1)-CLS/Benchmark 0.9891 1.0325 1.0230 1.1233 1.0399 1.1033 RGRMSE INMA(1)-YW/Benchmark 0.9654 1.0118 0.9987 1.0970 1.0152 1.0773 M.Mohammadipour, 2009, Chapter 9 221 Looking at the estimated parameters of the INMA(1) process (?,?) reveals that in general, ? is close to zero (the average is 0.0374 and 79.94 percent are between 0 and 0.05) and ? is between 2 and 3 (the average is 2.7357 and 43.55 percent are between 2 and 3). The simulation results for the corresponding parameters (similar values of ?, ?, and ?) show that YW yields slightly better results than CLS and also both CLS and YW results are close to SBA and SBJ with smoothing parameter 0.2 (with INMA slightly better in terms of MSE). The results of Table 9-17 show that YW forecasts are better than CLS in terms of MASE (only by 0.1 percent) but worse in terms of MSE (only by 0.5 percent). The best INMA(1) is better than the best benchmark by only 0.4 percent using MSE and worse than the best benchmark by 1.8 percent using MASE. Finally, the results of identification show that 3.60 percent of 1,943 series are identified as INARMA(1,1). We assume that the order of these 70 series is taken to be (1,1) and the INARMA(1,1) forecasting method is used. We also investigate the effect of using different estimation methods (CLS and YW) on the forecasting accuracy. The results are given in Table 9-19 and Table 9-20. Table ?9-19 Only INARMA(1,1) series for all points in time (3000 series) Accuracy measure Forecasting method Croston 0.2 Crosto n 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 INMA(1) CLS INMA(1) YW ME -0.1340 -0.1141 0.1556 0.6049 0.1878 0.8446 -0.2016 -0.0997 MSE 4.7747 5.3016 4.7204 5.2908 4.7275 5.6017 5.0449 5.8052 MASE 0.8853 0.9369 0.8695 0.9076 0.8690 0.9192 0.9178 0.9849 PB of MASE INMA(1)-CLS/Benchmark 0.4607 0.5226 0.4452 0.4821 0.4512 0.5036 PB of MASE INMA(1)-YW/Benchmark 0.4417 0.4881 0.4369 0.4702 0.4429 0.4905 RGRMSE INMA(1)-CLS/Benchmark 1.0854 1.0284 1.1681 1.0737 1.1793 1.1495 RGRMSE INMA(1)-YW/Benchmark 0.8169 0.7752 0.8747 0.7943 0.8910 0.8146 In general, the estimated autoregressive parameter of an INARMA(1,1) process is in the range 0.1 ? ? Equation 10-4 where ?? ?? is the cross-covariance at lag ? given by Equation 10-3. Therefore, as the second finding of this research, the autocorrelation function of an INARMA(p,q) process is as follows: ??=? ?1???1 +?2???2 +?+??????+???0 ??+??+1?1 ??+?+?????? ?? ?0 ??? ?1???1 +?2???2 +?+?????? ?>? ? Equation 10-5 where ?0 is the unconditional variance of the process and is given by Equation 10-2. It is shown that if ?= 0, the Equation 10-5 reduces to the ACF of an INAR(p) process given by Du and Li (1991). Also if ?= 0, the Equation 10-5 provides the ACF of an INMA(q) process which is given by Br?nn?s and Hall (2001) (see section 3.3.8.2). It is also shown is section 4.3.1 that the ACF structure of the INARMA(p,q) process is analogous to that of an ARMA(p,q) process. Also, based on the discussion in section 4.3.2.3, the partial autocorrelation function of an INARMA(p,q) process is the analogue of that of an ARMA process, is infinite and behaves like the PACF of a pure integer moving average process. M.Mohammadipour, 2009, Chapter 10 246 10.3.3 The YW Estimators of an INARMA(1,1) Model Based on Equation 10-2 and Equation 10-5, we have found the YW estimators for an INARMA(1,1) process to be: ?= ?2 ?1 = ? ??????(???2??) ? ?=3 ? ??????(???1??) ? ?=2 Equation 10-6 ?= ?1 +??(???1) ?1?1 + 3???1???2?2 Equation 10-7 where ?? is the estimate of the autocorrelation at lag ?, ??. The innovation parameter is then estimated from the expected value of the process ?= 1?? 1 +? ? ?? ? ?=1 ? Equation 10-8 We have also derived the YW estimators for an INARMA(2,2) process for presentational purposes (see section 5.8). 10.3.4 Lead Time Forecasting of an INARMA(p,q) Model It is shown in chapter 6 that aggregation of an INARMA(p,q) process over a lead time results in an INARMA(p,q) process. The aggregated process has the same autoregressive and moving average parameters but a different innovation parameter. When the innovations of the original process are ??~???(?) , the innovations of the aggregated process are ??~???(??+ 1??). It is also found that the aggregated process can be written in terms of the last ? observations as follows: M.Mohammadipour, 2009, Chapter 10 247 ? ??+? ?+1 ?=1 =? ? ?? 1 ??? ?? 1 ?=1 ?+1 ?=1 +? ? ?? 2 ????1 ?? 2 ?=1 ?+1 ?=1 +? +? ? ?? ??????+1 ?? ? ?=1 ?+1 ?=1 +? ? ?? ?+1???+?? ?? ?+1 ?=1 ?+1 ?=1 Equation 10-9 where all the parameters are defined in Table 6-4. The above result is then used to find the conditional expected value of the aggregated process which is: ??? ??+? ?+1 ?=1 |????+1,?,???1 ,???=?? ? ?? 1 ?? 1 ?=1 ?+1 ?=1 ???+?? ? ?? 2 ?? 2 ?=1 ?+1 ?=1 ????1 +? +?? ? ?? ? ?? ? ?=1 ?+1 ?=1 ?????+1 +?? ? ?? ?+1 ?? ?+1 ?=1 ?+1 ?=1 ?? Equation 10-10 Based on the above results, the lead time forecasts for the specific INARMA processes used in this research are derived as follows: ??? ??+? ?+1 ?=1 |???= ? ? ? ? ? ? ? ? ? (?+ 1)? for INARMA(0,0) ?(1???+1) 1?? ??+ ? 1?? ???+ 1??? ?? ?+1 ?=1 ? for INAR(1) ??+ 1?(1 +?)? for INMA(1) ?(1???+1) 1?? ??+ ?(1 +?) 1?? ???+ 1??? ?? ?+1 ?=1 ? for INARMA(1,1) ? Equation 10-11 10.4 Conclusions from the Simulation Part of the Thesis In this section, the main findings of the simulation part of this research are summarized. Four INARMA models are used in this thesis which are: INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1). The range of parameters chosen for these processes is shown in Table 7-1. M.Mohammadipour, 2009, Chapter 10 248 10.4.1 The Performance of Different Estimation Methods The performance of YW, CLS, and CML (only for INAR(1) process) is compared. The simulation results show that, for an INAR(1) process, the MSE of estimates produced by CML is generally less than that of YW and CLS. However, the accuracy of forecasts produce by CML is not very far from those by YW and CLS. The YW and CLS estimators are close when the number of observations is high. But, for small samples, the difference is high when the autoregressive parameter is high. In such cases, CLS produces much better estimates for both ? and ? in terms of MSE than YW. On the other hand, for small values of ?, YW results in better estimates. This is also true for the accuracy of forecasts produced by these estimation methods. For an INMA(1) process, for small number of observations, CLS generally has smaller MSEs than YW except for very high values of ?. For large number of observations, YW has smaller MSEs than CLS for high values of ?. However, this does not have a great effect on the accuracy of forecasts produced by each method. The MSE of ? for both YW and CLS estimates increases with an increase in ? but the same is not necessarily true for the MSE of ?. Finally, for an INARMA(1,1) process, CLS generally produces better estimates especially when the number of observations is small and the autoregressive parameter is high. This is also true for the accuracy of forecasts produced by CLS compared to those by YW. The results of three-step and six-step ahead forecasts show that the forecasts based on YW and CLS estimation methods are generally very close for all of the three INARMA processes, but YW-based results are slightly better in many cases. 10.4.2 The Croston-SBA Categorization Syntetos et al. (2005) provide a categorization scheme for Croston and SBA based on MSE to establish the areas that each method should be used over the other. They use the squared coefficient of variation (??2) of demand size and the average inter- demand interval (?) to identify different regions shown in Figure 8-1. Because this M.Mohammadipour, 2009, Chapter 10 249 categorization is based on the assumption that demand occurs as an i.i.d. Bernoulli process, it is worth testing if it also holds for an i.i.d. Poisson (an INARMA(0,0)) process. The simulation results show that not only the Croston-SBA categorization based on MSE holds for i.i.d. Poisson demand, but it also generally holds using MASE. The simulation results also show that when the number of observation increases, the advantage of SBA over Croston decreases. This is also shown by direct mathematical calculations in section 8.5. MSESBA ?MSECroston ???1? ? 2 ? 2 ?1?? ?+ 2(1??)2?+1 2?? ?? (??1)2?2 ?4 + ?2 ?2 ? Equation 10-12 Based on the Equation 10-12, when the smoothing parameter is small (?= 0.2), the above coefficient decreases when ? increases; therefore the difference between MSE of Croston and SBA also decreases. However, because the above coefficient reaches a limit of ? ?2 4 ???? ? 2?? ?, the advantage of SBA over Croston does not change noticeably when the number of observations is high. On the other hand, when the smoothing parameter is high (?= 0.5), the difference between the MSE of Croston and SBA changes little with changes in ?. Although the Croston-SBA categorization is for i.i.d. demand, we have also tested it when demand is produced by an INAR(1), INMA(1) or an INARMA(1,1) process. The results show that the Croston-SBA categorization generally holds for all of these processes. 10.4.3 Identification in INARMA Models We test the two identification approaches mentioned in section 10.2. The simulation results show that when the data is produced by an INARMA(0,0) process, the two- stage method produces better results than the one-stage method. This is true in terms of the percentage of time that the model is identified correctly and also in terms of the accuracy of forecasts. However, when data is produced by INAR(1), INMA(1) or M.Mohammadipour, 2009, Chapter 10 250 INARMA(1,1) processes, the one-stage method outperforms the two-stage method. For INAR(1) and INARMA(1,1) processes, when the autoregressive parameter is low, the process is misidentified in most cases for both identification methods. But when the autoregressive parameter is high, the performance of both identification methods improves. Obviously, when more observations are available, the percentage of correct identification increases for both methods. The INMA(1) process is misidentified in most of the cases. However, the results show that it does not affect the forecasting accuracy to a great extent. In general, the performance of both identification methods improves for higher moving average parameters and longer length of history. The simulation results also show that, for INAR(1) and INARMA(1,1) processes, misidentification has a great impact on the accuracy of forecasts when the autoregressive parameter is high. When the autoregressive parameter is small, or the process is an INMA(1) process, the effect of misidentification on forecasting accuracy is small. We also test the case that the identification step is ignored and an INAR(1) or an INARMA(1,1) method is used for forecasting. The results show that when the number of observations is small, this approach produces better forecasts for INAR(1) and INMA(1) series. The results also show that the all-INAR(1) approach produces better forecasts than the all-INARMA(1,1) approach for all four INARMA series. 10.4.4 Comparing INARMA with the Benchmark Methods The results show that when the order of the INARMA model is known, the INARMA method almost always produces the lowest MSE when all points in time is considered. When only issue points are considered, the INARMA forecasts are biased and therefore are not always better than the benchmark methods. For INARMA(0,0) and INMA(1) processes, the improvement over benchmarks is not considerable. However, for the INAR(1) and INARMA(1,1) processes, the improvement is considerable when the autoregressive parameter is high. This is true M.Mohammadipour, 2009, Chapter 10 251 for both MSE and MASE of forecasts. The degree of improvement (by using INARMA over the benchmark methods) for the ?-step ahead forecasts does not change for INARMA(0,0) process. But for an INMA(1) process, the performance of INARMA compared to benchmark methods is improved for h-step ahead forecasts compared to one-step ahead forecasts. For INAR(1) and INARMA(1,1) processes, the performance of INARMA over the benchmark methods is improved compared to the one-step ahead case when the autoregressive parameter is low. But when the autoregressive parameter is high, the fact that the forecasts converge to the mean of the process results in poor forecasts compared to the one-step ahead case. All of the above results were for the case that the order of the INARMA model is known. This is obviously not true in practice and the INARMA model needs to be identified. The results of identification show that treating all as an INAR(1) model is a promising approach especially for high autoregressive parameters and short length of history. Therefore, we compare an all-INAR(1) method with the benchmark methods. The results show that for INARMA(0,0) and INMA(1) processes, the benchmark methods outperform INARMA especially for more sparse demand. For INARMA(1,1) series, when the autoregressive parameter is high, INARMA is considerably better than the benchmark methods in terms of MSE and MASE. Obviously, the results for INAR(1) series are the same as the case that the order is known. We also compare the lead time forecasts produced by an all-INAR(1) method with the benchmark methods. The results show that, for INARMA(0,0) and INMA(1) series, the all-INAR(1) forecasts are better than the best benchmark in most of the cases. For INAR(1) and INARMA(1,1) series, when the autoregressive parameter is high, the improvement of INARMA over the best benchmark is narrow for short length of history. However with more observations, the improvement also increases. For small autoregressive parameters, the INAR(1) method always outperforms the benchmark methods. Again, the improvement increases with an increase in the length of history. M.Mohammadipour, 2009, Chapter 10 252 10.5 Conclusions from the Empirical Part of the Thesis The simulation analysis of this research compared the accuracy of forecasts (in terms of MSE and MASE) by an INARMA method with those produced by benchmark methods when the data is generated by an INARMA model. Empirical analysis is performed to validate the theoretical and simulation results for real data. Two data sets used in this research are the Royal Air Force (RAF) individual demand histories of 16,000 SKUs over a period of 6 years (monthly observations) and demand history of 3,000 SKUs from the automotive industry over a period of 2 years (24 months). Both data sets are filtered to eliminate lumpy series, since the INARMA models with Poisson marginal distribution are not appropriate for lumpy demand. The PB of MASE and the RGRMSE are used to measure the forecast error in addition to ME, MSE and MASE. 10.5.1 Identification in INARMA Models The two identification methods (two-stage and one-stage) are compared. The empirical results for both 16,000 and 3,000 series show that the two-stage method produces more accurate forecasts. This is not surprising because the identification shows that the majority of series in both data sets are identified as INARMA(0,0) and the simulation results show that for INARMA(0,0) series the two-stage method is better than the one-stage method. The empirical results also confirm the simulation results that ignoring the identification step, and forecasting with an INARMA(1,1) or an INAR(1) process, is a promising approach. It is also confirmed that the latter outperforms the former. 10.5.2 The Performance of Different Estimation Methods After the appropriate INARMA model is identified for all the series in each data set, different INARMA series are separated and forecasted by the corresponding INARMA method. The performance of different estimation methods in terms of the accuracy of forecasts are tested for each INARMA model. M.Mohammadipour, 2009, Chapter 10 253 The empirical results agree with the simulation results for all three INARMA models. For INAR(1) series (with ? around 0.1 and ? around 2), the YW produces better forecasts than CLS and CML agreeing with the simulation results for similar parameters. For INMA(1) series, YW produces slightly better results than CLS. However, for INARMA(1,1) series, CLS outperforms YW, which again confirms the simulation results. 10.5.3 Comparing INARMA with the Benchmark Methods The forecast accuracy of INARMA (based on all identification methods) is compared to that of the benchmark methods for both 16,000 and 3,000 series. The results show that, for the former data set, there is an improvement by using INARMA methods for one-step ahead forecasts for all accuracy measures. The improvement is narrow for 3,000 series which is expected because of the short length of history and low autocorrelation. This slight improvement is based on MSE and, in fact, the MASE of INARMA is worse than the benchmarks for 3,000 series. The accuracy of ?-step ahead INARMA forecasts are even better than the benchmarks compared to the one-step ahead forecasts. This is true for both data sets. The lead time forecasts are obtained based on the conditional expected value of the aggregated INARMA process and also from the cumulative ?-step ahead forecasts. The empirical results show that for INAR(1) series of both data sets, both of these methods produce considerably better forecasts than the benchmark methods with the former outperforming the latter. The same is true for INMA(1) series of both data sets but not for INARMA(1,1) series. For INARMA(1,1) series, the cumulative ?- step ahead forecasts are better than those based on conditional expected value of the aggregated process. It should be borne in mind that the number of INARMA(1,1) for both data sets are very small. 10.5.4 The Problem with MASE The empirical results show that when the data series is highly intermittent (which is M.Mohammadipour, 2009, Chapter 10 254 the case for most of the 16,000 series), the MASE of all forecasting methods is very high. This would suggest that all methods are worse than na?ve. Because the data series contain many zeros in the estimation period, the error of na?ve in most of the time periods is zero. Therefore the in-sample MAE is very small and the MASE is very large. This is true when all points in time are considered. However, for the case of issue points, because it is more likely that a nonzero demand is followed with a zero demand, the absolute error of na?ve and therefore the in-sample MAE is large. As a result, the MASE of the forecasting methods is smaller compared to the all points in time case. The empirical results confirm this. As a result, for highly intermittent data, MASE does not provide reliable results. 10.6 Practical and Software implications In this thesis we have applied an INARMA method to model and forecast non-erratic intermittent demand (see the classification by Boylan et al., 2008). Four models have been assumed to include autoregressive, moving average, mixed models and also an i.i.d. Poisson process. All simulation and empirical results are based on the following assumptions: ? The intermittent demand can be modelled with either an INARMA(0,0), INAR(1), INMA(1) or an INARMA(1,1) process with Poisson marginal ? The benchmark methods to compete against INARMA method are: Croston, SBA and SBJ ? The forecast accuracy measures are: ME, MSE, MASE, RGRMSE, and PB, with the last two only used for empirical analysis It has been shown that INARMA performs best when data has high autocorrelation. Even with low autocorrelation INARMA outperforms the benchmarks when the length of history is large. However, we cannot claim that when the above assumptions are violated, these results still hold. For example, the empirical data of this research were filtered to comply with the first assumption. Therefore, in order to use the INARMA method, the distribution that fits the demand M.Mohammadipour, 2009, Chapter 10 255 data of the organization should first be found. All discrete self-decomposable distributions can be used as marginal distributions of INARMA models. This includes: Poisson, generalized Poisson, and negative binomial distributions. The generalized Poisson allows for both under and overdispersion (Br?nn?s, 1994), and the negative binomial allows for overdispersion (McCabe and Martin, 2005). The results of this research suggest that although identification of the autoregressive and moving average orders of the INARMA models results in more accurate forecasts, using simple models such as an INAR(1) model also produces good forecasts. This is an especially useful method when the length of history is not long enough for identification. However, it should be mentioned that the simulation results of this research were based on limited INARMA models and the empirical data did not support higher order models. Therefore, for higher order models, the performance of using a simple model instead of identification needs to be tested first through simulation. The YW and CLS estimation methods are not based on distributional assumptions and therefore can be used for all cases. However, the maximum likelihood estimators for the corresponding distribution should be obtained. The results of this research can be used, at least for the specific INARMA models and parameters, as a guide to when one estimation method should be used over another. Although the lead time forecasts in this thesis are restricted to Poisson, the findings can be easily amended for other distributions. The conventional ARIMA models are offered by standard forecasting programmes such as Autobox (Automatic Forecasting Systems) and Forecast Pro (Business Forecast Systems). These programmes provide automated identification, estimation and forecasting for ARIMA models. For instance, Forecast Pro uses the Bayesian information criterion (BIC) along with some other rules for identification and unconditional least squares for estimation (Goodrich, 2000). Autobox matches the SACF with theoretical ACF for some starting ARIMA models and then uses AIC to select the best model (Reilly, 2000). Although INARMA models have had applications in many areas (see section 3.3.9), there is currently no software based on these models. This research suggested a new M.Mohammadipour, 2009, Chapter 10 256 application area for INARMA models in intermittent demand forecasting. The improvement in forecast accuracy by using INARMA over the benchmark methods such as Croston (which is offered in some forecasting programmes such as Forecast Pro), also urges the need to develop forecasting software for these models. In doing so, this research can be considered a starting point. We have addressed issues of whether AIC produces satisfactory results, which estimation method is best in what case, and ?-step ahead and lead time forecasting for a limited number of INARMA models. 10.7 Limitations and Further Research Throughout this research we have made assumptions that can be relaxed in future studies. Although most of the theoretical findings are not based on any marginal distribution, for simulation and empirical analyses we restricted the research to a specific distribution. As previously discussed in section 7.2, the Poisson distribution was selected due to its interesting properties. However, it does not allow for overdispersion in data and in such cases other distributions should be assumed. For example, Alzaid and Al-Osh (1993) show that the Generalized Poisson INAR(1) process has more variability than PoINAR(1) because its variance is twice that of PoINAR(1). This could be useful for modelling moderately lumpy demand. Other adaptations of INARMA models to take into account non-stationarity, seasonality and trend can also be used. For example the signed binomial thinning models introduced by Kim and Park (2008) allow for negative values and negative autocorrelations and also use the differencing operator to remove the trend and seasonality. As mentioned in section 7.3.1, the simulation and empirical results of this research have been based on INARMA models with ?,??1. A natural extension would be to use higher order models. As seen in chapter 9, our empirical data did not support such models; however, other data sets may be better fitted by higher order INARMA models. M.Mohammadipour, 2009, Chapter 10 257 This research has only focused on aggregation of INARMA models over lead time. The temporal and cross-sectional aggregation in INARMA models were briefly reviewed in chapter 3. A problem is identified with the result of temporal aggregation of an INAR(1) process provided by Br?nn?s et al. (2002). More studies should be carried out in both temporal and cross-sectional aggregation fields. As mentioned in section 4.5, the complexity of the likelihood function has restricted the use of penalty functions for INARMA models. We have used the AIC based on the likelihood function of ARIMA models and shown that the results are satisfactory. However, there is still a need for finding the AIC based on INARMA models. As confirmed in chapters 8 and 9, the INARMA forecasts are biased when only issue points are considered. The estimates, and as a result, forecasts, can be revised to reduce such bias. For example, the CLS of an INAR(1) process is based on the following criterion: ?????=? [??????????1?] 2 ? ?=1 Equation 10-13 where ?= (?,?)?. Now, when only issue points are considered, the estimates of the parameters will be updated after observing a positive demand. Therefore, in Equation 10-13 the last observation which is ?? is definitely positive and its conditional expected value should be ???????1,??> 0? instead of ???????1?. As a result, the new least squares criterion will be: ?? ????=? [??????????1?] 2 ??1 ?=1 + [??????????1,??> 0?] 2 Equation 10-14 The revised CLS estimates of parameters can be obtained by minimization of the above criterion. Numerical methods are needed to find these estimates. The new estimates can be compared to CLS estimates in terms of their impact on forecast accuracy when only issue points are considered. Other estimation methods that allow for such revisions such as maximum likelihood could also be considered. Another limitation of this study is that we have used the conditional expectation to produce forecasts. This enabled us to compare the MMSE INARMA forecasts to M.Mohammadipour, 2009, Chapter 10 258 point forecasts from benchmark methods. The natural next step is to forecast the whole distribution instead. This has been done in a number of studies in the INARMA literature (Freeland and McCabe, 2004b; McCabe and Martin, 2005; Bu and McCabe, 2008). This will then enable us to compare INARMA method with bootstrapping methods in an IDF context. Incorporating explanatory variables in INARMA models has been the subject of some studies (Br?nn?s, 1995; Br?nn?s and Quoreshi, 2004). As discussed in section 2.3.3, causal models for IDF have not yet been well developed in the literature and the integration of these models with INARMA models would be an interesting line of research. The benchmark methods of this research were Croston, SBA and SBJ. We have used two arbitrary values for the smoothing parameter of these methods. This could be replaced by the optimum smoothing parameter. Also, different smoothing parameters could be used for updating demand size and inter-arrival time. Although this study has only focused on forecasting and not inventory control, it is expected that the improvement in the mean (forecasts) would translate to better percentile estimates and better inventory results. Obviously, this could be tested in future studies to find whether using INARMA method would result in better inventory measures such as service level and inventory level that have been suggested by Teunter and Duncan (2009). Different inventory models need different estimation of parameters, i.e. mean, percentiles, or estimates of demand sizes and inter-demand invervals. Syntetos et al. (2008) develop a modified periodic order-up-to-level inventory policy for intermittent demand which relies upon demand sizes and inter-demand intervals. Depending on what inventory model is applied, INARMA methods can provide all of these estimates. For example, in the case where demand follows an INAR(1) model: ??demand size?= ?/(1??) 1????/(1??) Equation 10-15 ??demand interval?= 1?? ? Equation 10-16 M.Mohammadipour, 2009, Chapter 10 259 Finally, the empirical results of this research have been restricted to two data sets. More empirical analyses can be done to further assess the sensitivity of results to the length of history and the level of intermittence. M.Mohammadipour, 2009, References 260 References Adya, M., F. Collopy, J. S. Armstrong and M. Kennedy (2001). Automatic identification of time series features for rule-based forecasting. International Journal of Forecasting 17(2): 143-157. Aggoun, L., L. Benkherouf and L. Tadj (1997). A hidden Markov model for an inventory system with perishable items. Journal of Applied Mathematics and Stochastic Analysis 10(4): 423-430. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. B. N. Petrov and F. Csaki (Eds.), Proceedings of the second International Symposium on Information Theory, Akad?miai Kaid?, Budapest. Al-Osh, M. A. and A. A. Alzaid (1987). First order integer-valued autoregressive (INAR(1)) processes. Journal of Time Series Analysis 8(3): 261-275. Al-Osh, M. A. and A. A. Alzaid (1988). Integer-valued moving average (INMA) process. Statistical Papers 29(4): 281-300. Aly, E.-E. A. A. and N. Bouzar (1994). On some integer-valued autoregressive moving average models. Journal of Multivariate Analysis 50(1): 132-151. Aly, E.-E. A. A. and N. Bouzar (2005). Stationary solutions for integer-valued autoregressive processes. International Journal of Mathematics and Mathematical Sciences 2005(1): 1-18. Alzaid, A. A. and M. Al-Osh (1988). First-order integer-valued autoregressive (INAR(1)) process: distributional and regression properties. Statistica Neerlandica 42(1): 53-61. Alzaid, A. A. and M. Al-Osh (1990). An integer-valued pth-order autoregressive structure (INAR(p)) process. Journal of Applied Probability 27(2): 314-324. Alzaid, A. A. and M. Al-Osh (1993). Some autoregressive moving average processes with generalized Poisson marginal distributions. Annals of the Institute of Statistical Mathematics 45(2): 223-232. Amemiya, T. and R. Y. Wu (1972). The effect of aggregation on prediction in the autoregressive model. Journal of the American Statistical Association 67(339): 628- 632. Berglund, E. and K. Br?nn?s (2001). Plants' entry and exit in Swedish municipalities. The Annals of Regional Science 35(3): 431-448. Blundell, R., R. Griffith and F. Windmeijer (2002). Individual effects and dynamics in count data models. Journal of Econometrics 108(1): 113-131. M.Mohammadipour, 2009, References 261 B?ckenholt, U. (1999). Mixed INAR(1) Poisson regression models: Analyzing heterogeneity and serial dependencies in longitudinal count data. Journal of Econometrics 89(1-2): 317-338. Bookbinder, J. H. and A. E. Lordahl (1989). Estimation of inventory re-order levels using the bootstrap statistical procedure. IIE Transactions 21(4): 302-312. Bowerman, B. L., R. T. O'Connell and A. B. Koehler (2005). Forecasting, time series, and regression, 4th ed., Thomson Brooks/Cole: Belmont, CA. Box, G. E. P. and G. M. Jenkins (1970). Time series analysis: Forecasting and control, Holden-Day: San Francisco. Box, G. E. P., G. M. Jenkins and G. C. Reinsel (1994). Time series analysis: Forecasting and control, 3rd ed., Prentice Hall: New Jersey. Boylan, J. E. and A. A. Syntetos (2006). Accuracy and accuracy implication metrics for intermittent demand. FORESIGHT: International Journal of Applied Forecasting (4): 39-42. Boylan, J. E. and A. A. Syntetos (2007). The accuracy of a modified Croston procedure. International Journal of Production Economics 107(2): 511-517. Boylan, J. E. and A. A. Syntetos (2008). Forecasting for inventory management of service parts In Complex system maintenance handbook. K. A. H. Kobbacy and D. N. P. Murthy (eds.), Springer: London, 479-508. Boylan, J. E., A. A. Syntetos and G. C. Karakostas (2008). Classification for forecasting and stock control: a case study. Journal of the Operational Research Society 59(4): 473-481. Brandt, P. T., J. T. Williams, B. O. Fordham and B. Pollins (2000). Dynamic modeling for persistent event-count time series. American Journal of Political Science 44(4): 823-843. Br?nn?s, K. (1994). Estimation and testing in integer-valued AR(1) models. Ume? Economic Studies 335, Ume? University. Br?nn?s, K. (1995). Explanatory variables in the AR(1) count data model. Ume? Economic Studies 381, Ume? University. Br?nn?s, K. and A. Hall (2001). Estimation in integer-valued moving average models. Applied Stochastic Models in Business and Industry 17(3): 277-291. Br?nn?s, K. and J. Hellstr?m (2001). Generalized integer-valued autoregression. Econometric Reviews 20(4): 425-443. Br?nn?s, K., J. Hellstr?m and J. Nordstr?m (2002). A new approach to modelling and forecasting monthly guest nights in hotels. International Journal of Forecasting 18(1): 19-30. M.Mohammadipour, 2009, References 262 Br?nn?s, K. and P. Johansson (1994). Time series count data regression. Communications in Statistics Theory and Methods 23(10): 2907-2925. Br?nn?s, K. and S. Quoreshi (2004). Integer-valued moving average modelling of the number of transactions in stocks. Ume? Economic Studies 637, Ume? University. Brewer, K. R. W. (1973). Some consequences of temporal aggregation and systematic sampling for ARMA and ARMAX models. Journal of Econometrics 1(2): 133-154. Brockwell, P. J. and R. A. Davis (1996). Introduction to time series and forecasting, Springer-Verlag: New York. Brown, R. G. (1959). Statistical forecasting for inventory control, McGraw-Hill: New York. Bu, R. (2006). Essays in financial econometrics and time series analysis. PhD thesis: The University of Liverpool, UK. Bu, R. and B. McCabe (2008). Model selection, estimation and forecasting in INAR(p) models: A likelihood-based Markov Chain approach. International Journal of Forecasting 24(1): 151-162. Bu, R., B. McCabe and K. Hadri (2008). Maximum likelihood estimation of higher- order integer-valued autoregressive processes. Journal of Time Series Analysis 29(6): 973-994. Buishand, T. A. (1978). The binary DARMA(1,1) process as a model for wet-dry sequences. Technical note 78-01, Department of Mathematics, Agricultural University Wageningen. Cardinal, M., R. Roy and J. Lambert (1999). On the application of integer-valued time series models for the analysis of disease incidence. Statistics in Medicine 18(15): 2025-2039. Chang, T. J., M. L. Kavvas and J. W. Delleur (1984). Daily precipitation modeling by discrete autoregressive moving average processes. Water Resources Research 20(5): 565-580. Chatfield, C. and D. L. Prothero (1973). Box-Jenkins seasonal forecasting: Problems in a case-study. Journal of the Royal Statistical Society, Series A 136(3): 295-336. Collopy, F. and J. S. Armstrong (1992). Rule-based forecasting: Development and validation of an expert systems approach to combining time series extrapolations. Management Science 38(10): 1394-1414. Cox, D. R. (1961). Prediction by exponentially weighted moving averages and related methods. Journal of the Royal Statistical Society. Series B 23(2): 414-422. M.Mohammadipour, 2009, References 263 Croston, J. D. (1972). Forecasting and stock control for intermittent demands. Operational Research Quarterly 23(3): 289?303. Drost, F. C., R. v. d. Akker and B. J. M. Werker (2008). Local Asymptotic Normality and efficient estimation for INAR(p) models Journal of Time Series Analysis 29(5): 783-801. Drost, F. C. and T. E. Nijman (1993). Temporal aggregation of GARCH processes. Econometrica 61(4): 909-927. Du, J. G. and Y. Li (1991). The integer-valued autoregressive (INAR(p)) model. Journal of Time Series Analysis 12(2): 129-142. Durbin, J. and S. J. Koopman (2000). Time series analysis of non-Gaussian observations based on state space models from both classical and Bayesian perspectives. Journal of the Royal Statistical Society, Series B 62(1): 3-56. Eaves, A. H. C. and B. G. Kingsman (2004). Forecasting for the ordering and stock- holding of spare parts. Journal of the Operational Research Society 55(4): 431-437. Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Annals of Statistics 7(1): 1-26. Enciso-Mora, V., P. Neal and T. S. Rao (2008). Efficient order selection algorithms for integer-valued ARMA processes. Journal of Time Series Analysis 30(1): 1-18. Fildes, R. (1992). The evaluation of extrapolative forecasting methods. International Journal of Forecasting 8(1): 81-98. Franke, J. and T. Seligmann (1993). Conditional maximum-likelihood estimates for INAR(1) processes and their applications to modelling epileptic seizure counts. In Developments in Time Series Analysis. T. S. Rao (Ed.), Chapman and Hall: London, 310-330. Franke, J. and T. Subba Rao (1995). Multivariate first order integer valued autoregressions. Berichte der Arbeitsgruppe Technomathematik, Bericht Nr.95, Universit?t Kaiserslautern. Freeland, K. (1998). Statistical analysis of discrete time series with application to the analysis of workers' compensation claims data. PhD thesis: University of British Columbia. Freeland, R. K. and B. P. M. McCabe (2004a). Analysis of low count time series data by Poisson autoregression. Journal of Time Series Analysis 25(5): 701-722. Freeland, R. K. and B. P. M. McCabe (2004b). Forecasting discrete valued low count time series. International Journal of Forecasting 20(3): 427-434. M.Mohammadipour, 2009, References 264 Freeland, R. K. and B. P. M. McCabe (2005). Asymptotic properties of CLS estimators in the Poisson AR(1) model. Statistics and Probability letters 73(2): 147- 153. Gardner, E. S. and A. B. Koehler (2005). Comments on a patented bootstrapping method for forecasting intermittent demand. International Journal of Forecasting 21(3): 617-618. Ghobbar, A. A. and C. H. Friend (2002). Sources of intermittent demand for aircraft spare parts within airline operations. Journal of Air Transport Management 8(4): 221-231. Ghobbar, A. A. and C. H. Friend (2003). Evaluation of forecasting methods for intermittent parts demand in the field of aviation: a predictive model. Computers and Operations Research 30(14): 2097-2114. Goodrich, R. L. (2000). The Forecast Pro methodology. International Journal of Forecasting 16(4): 533-535 Goodwin, P. and R. Lawton (1999). On the asymmetry of the symmetric MAPE. International Journal of Forecasting 15(4): 405-408. Gooijer, J. G. and R. J. Hyndman (2006). 25 years of time series forecasting. International Journal of Forecasting 22(3): 443-473. Gourieroux, C. and J. Jasiak (2004). Heterogeneous INAR(1) model with application to car insurance. Insurance: Mathematics and Economics 34(2): 177-192. Graves, S. C. (1999). A single-item inventory model for a nonstationary demand process. Manufacturing & Service Operations Management 1(1): 50-61. Grunwald, G. K., R. J. Hyndman, L. Tedesco and R. L. Tweedie (2000). Non- Gaussian conditional linear AR(1) models. The Australian & New Zealand Journal of Statistics 42(4): 479-495. Hall, B. H., Z. Griliches and J. A. Hausman (1986). Patents and R and D: is there a lag? International Economic Review 27(2): 265-283. Hamilton, J. D. (1994). Time series analysis, Princeton University Press: Princeton, New Jersey. Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50(4): 1029-1054. Harvey, A. C. and C. Fernandes (1989). Time series models for count data or qualitative observations. Journal of Business & Economic Statistics 7(4): 407-417. Harvey, A. C. and R. G. Pierse (1984). Estimating missing observations in economic time series. Journal of the American Statistical Association 79(385): 125-131. M.Mohammadipour, 2009, References 265 Hua, Z. S., B. Zhang, J. Yang and D. S. Tan (2007). A new approach of forecasting intermittent demand for spare parts inventories in the process industries. Journal of the Operational Research Society 58(1): 52-61. Hurvich, C. M. and C. L. Tsai (1989). Regression and time series model selection in small samples. Biometrika 76(2): 297-307. Hurvich, C. M. and C. L. Tsai (1995). Model selection for extended quasi-likelihood models in small samples. Biometrics 51(3): 1077-1084. Hyndman, R. J. (2006). Another look at forecast accuracy metrics for intermittent demand. FORESIGHT: International Journal of Applied Forecasting (4): 43-46. Hyndman, R. J. and A. B. Koehler (2006). Another look at measures of forecast accuracy. International Journal of Forecasting 22(4): 679-688. Jacobs, P. A. and P. A. W. Lewis (1978a). Discrete time series generated by mixtures I: Correlational and runs properties. Journal of the Royal Statistical Society. Series B 40(1): 94-105. Jacobs, P. A. and P. A. W. Lewis (1978b). Discrete time series generated by mixtures II: Asymptotic properties. Journal of the Royal Statistical Society. Series B 40(2): 222-228. Jacobs, P. A. and P. A. W. Lewis (1983). Stationary discrete autoregressive moving average time series generated by mixtures. Journal of Time Series Analysis 4(1): 19- 36. Johnston, F. R. and J. E. Boylan (1996). Forecasting for items with intermittent demand. Journal of the Operational Research Society 47(1): 113?121. Johnston, F. R., J. E. Boylan and E. A. Shale (2003). An examination of the size of orders from customers, their characterisation and the implications for the inventory control of slow moving items. Journal of the Operational Research Society 54(5): 833-837. Jung, R. C. and A. R. Tremayne (2003). Testing for serial dependence in time series models of counts. Journal of Time Series Analysis 24(1): 65-84. Jung, R. C. and A. R. Tremayne (2006a). Binomial thinning models for integer time series. Statistical Modelling 6(2): 81-96. Jung, R. C. and A. R. Tremayne (2006b). Coherent forecasting in integer time series models. International Journal of Forecasting 22(2): 223-238. Kang, I.-B. (2003). Multi-period forecasting using different models for different horizons: an application to U.S. economic time series data. International Journal of Forecasting 19(3): 387-400. M.Mohammadipour, 2009, References 266 Karlis, D. (2002). Overdispersed integer valued autoregressive models. COMPSTAT 2002, Berlin, Germany. Kim, H.-Y. and Y. Park (2008). A non-stationary integer-valued autoregressive model. Statistical Papers 49(3): 485-502. Kleijnen, J. P. C. and W. Groenendaal (1992). Simulation: a statistical perspective, John Wiley & Sons, Inc.: New York, NY, USA. Klimko, L. A. and P. I. Nelson (1978). On conditional least squares estimation for stochastic processes. Annals of Statistics 6(3): 629-642. Kolassa, S. and W. Sch?tz (2007). Advantages of the MAD/Mean ratio over the MAPE. FORESIGHT: The International Journal of Applied Forecasting (6): 40-43. Kostenko, A. and R. Hyndman (2006). A note on the categorization of demand patterns. Journal of the Operational Research Society 57(10): 1256-1257. Ksenija, C. (2006). Statistical analysis of wet and dry spells in Croatia by the binary DARMA(1,1) model. Croatian Meteorological Journal 41: 43-51. Kullback, S. and R. A. Leibler (1951). On information and sufficiency. The Annals of Mathematical Statistics 22(1): 79-86. Latour, A. (1998). Existence and stochastic structure of a non-negative integer- valued autoregressive process. Journal of Time Series Analysis 19(4): 439-455. Lawrence, A., A. Klimko and P. I. Nelson (1978). On conditional least squares estimation for stochastic processes. The Annals of Statistics 6(3): 629-642. Lee, H. L., K. C. So and C. S. Tang (2000). The value of information sharing in a two-level supply chain. Management Science 46(5): 626-643. Lev?n, E. and A. Segerstedt (2004). Inventory control with a modified Croston procedure and Erlang distribution. International Journal of Production Economics 90(3): 361-367. Ljung, G. M. and G. E. P. Box (1978). On a measure of lack of fit in time series models. Biometrika 65(2): 297-303. MacDonald, I. L. and W. Zucchini (1997). Hidden Markov and other models for discrete-valued time series, Chapman and Hall: London. Makridakis, S., A. Andersen, R. Carbone, R. Fildes, M. Hibon, R. Lewandowski, J. Newton, E. Parzen and R. Winkler (1982). The accuracy of extrapolation (time series) methods: Results of a forecasting competition. Journal of Forecasting 1(2): 111-153. Makridakis, S. and M. Hibon (1995). Evaluating accuracy (or error) measures. Working Paper 95/18/TM, INSEAD M.Mohammadipour, 2009, References 267 Makridakis, S. and M. Hibon (2000). The M3-competition: Results, conclusions and implications. International Journal of Forecasting 16(4): 451-476. Man, K. S. (2004). Linear prediction of temporal aggregates under model misspecification. International Journal of Forecasting 20(4): 659-670. Marcellino, M. (1999). Some consequences of temporal aggregation in empirical analysis. Journal of Business & Economic Statistics 17(1): 129-136. McCabe, B. P. M. and G. M. Martin (2005). Bayesian predictions of low count time series. International Journal of Forecasting 21(2): 315-330. McKenzie, E. (1985). Some simple models for discrete variate series. Water Resources Bulletin 21(4): 645-650. McKenzie, E. (1986). Autoregressive moving-average processes with negative binomial and geometric marginal distributions. Advances in Applied Probability 18(3): 679-705. McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts. Advances in Applied Probability 20(4): 822-835. McKenzie, E. (2003). Discrete variate time series. In Stochastic Processes: Modelling and Simulation. D. N. Shanbhag and C. R. Rao (Eds.), Elsevier: Amsterdam, 573-606. Mills, T. M. and E. Seneta (1989). Goodness-of-fit for a branching process with immigration using sample partial autocorrelations. Stochastic Processes and their Applications 33(1): 151-161. Neal, P. and T. S. Rao (2007). MCMC for integer-valued ARMA processes. Journal of Time Series Analysis 28(1): 92-110. Newbold, P. and C. W. J. Granger (1974). Experience with forecasting univariate time series and the combination of forecasts. Journal of the Royal Statistical Society. Series A 137(2): 131-165. Nijman, T. E. and F. C. Palm (1990). Predictive accuracy gain from disaggregate sampling in ARIMA models. Journal of Business & Economic Statistics 8(4): 405- 415. Ozaki, T. (1977). On the order determination of ARIMA models. Applied Statistics 26(3): 290-301. Park, Y. and C. W. Oh (1997). Some asymptotic properties in INAR(1) processes with Poisson marginals. Statistical Papers 38(3): 287-302. Patton, J. D. and R. J. Steele (2003). Service Parts Handbook, 2nd Edition, Solomon Pr: Rego Park, NY. M.Mohammadipour, 2009, References 268 Porras, E. and R. Dekker (2007). An inventory control system for spare parts at a refinery: An empirical comparison of different re-order point methods. European Journal of Operational Research 184(1): 101-132. Porras Musalem, E. (2005). Inventory theory in practice: Joint replenishments and spare parts control. PhD thesis: Erasmus University Rotterdam (Tinbergen Institute). Quddus, M. A. (2008). Time series count data models: An empirical application to traffic accidents. Accident Analysis & Prevention 40(5): 1732-1741 Quenouille, M. H. (1958). Discrete autoregressive schemes with varying time intervals. Metrika 1(1): 21-27. Rao, A. V. (1973). A comment on "Forecasting and stock control for intermittent demands". Operational Research Quarterly 24(4): 639-640. Reilly, D. (2000). The AUTOBOX system. International Journal of Forecasting 16(4): 531-533 Rengifo, E. W. (2005). Econometric analysis of financial count data and portfolio choice: A dynamic approach. PhD: Universit? Catholique de Louvain. Rudholm, N. (2001). Entry and the number of firms in the Swedish pharmaceuticals market. Review of Industrial Organization 19(3): 351-364. Salas, J. D., C.-h. Chung and B. Fernandez (2001). Relating autocorrelations and crossing rates of continuous and discrete valued hydrologic processes. ASCE Journal of Hydrologic Engineering 6(2): 109-118. Saunders, M., P. Lewis and A. Thornhill (2003). Research methods for business students, 3rd ed., Financial Times Prentice Hall: London. Schultz, C. L. (1987). Forecasting and inventory control for sporadic demand under periodic review. Journal of the Operational Research Society 38(5): 453-458. Segerstedt, A. (2000). Forecasting slow-moving items and ordinary items: a modification of?Croston?s?idea. Department of Business Administration and Social Science, Division of Industrial Logistics, Lule? University of Technology. Shale, E. A., J. E. Boylan and F. R. Johnston (2006). Forecasting for intermittent demand: the estimation of an unbiased average. Journal of the Operational Research Society 57(5): 588-592. Shenstone, L. and R. J. Hyndman (2005). Stochastic models underlying Croston's method for intermittent demand forecasting. Journal of Forecasting 24(6): 389-402. Silva, I. and M. E. Silva (2006). Asymptotic distribution of the Yule-Walker estimator for INAR(p) processes. Statistics & Probability Letters 76(15): 1655-1663. M.Mohammadipour, 2009, References 269 Silva, I., M. E. Silva, I. Pereira and N. Silva (2005). Replicated INAR(1) processes. Methodology and Computing in Applied Probability 7(4): 517-542. Silva, M. E. and V. L. Oliveira (2004). Difference equations for the higher-order moments and cumulants of the INAR(1) model. Journal of Time Series Analysis 25(3): 317-333. Silva, M. E. and V. L. Oliveira (2005). Difference equations for the higher order moments and cumulants of the INAR(p) model. Journal of Time Series Analysis 26(1): 17-36. Silver, E. A., D. F. Pyke and R. Peterson (1998). Inventory management and production planning and scheduling, 3rd ed., John Wiley & Sons: New York. Smith, J. (1979). A generalization of the Bayesian forecasting model. Journal of the Royal Statistical Society. Series B 41(3): 375-387. Snyder, R. (2002). Forecasting sales of slow and fast moving inventories. European Journal of Operational Research 140(3): 684-699. Snyder, R. D. and A. Beaumont (2007). A comparison of methods for forecasting demand for slow moving car parts. Monash Econometrics and Business Statistics Working Papers 15/07, Monash University. Sprott, D. A. (1983). Estimating the parameters of a convolution by maximum likelihood. Journal of the American Statistical Association 78(382): 457-460. Stuart, A. and J. K. Ord (1994). Kendall's advanced theory of statistics, 6th Edition, Edward Arnold: London. Sueutel, F. W. and K. van Harn (1979). Discrete analogues of self-decomposability and stability. Annals of Probability 7(5): 893-899. Syntetos, A. A. (2001). Forecasting for Intermittent Demand. PhD thesis: Buckinghamshire Chilterns University College, Brunel University, UK. Syntetos, A. A., M. Z. Babai, Y. Dallery and R. Teunter (2008). Periodic control of intermittent demand items: theory and empirical analysis. Journal of the Operational Research Society: doi:10.1057/palgrave.jors.2602593. Syntetos, A. A. and J. E. Boylan (2001). On the bias of intermittent demand estimates. International Journal of Production Economics 71(1-3): 457?466. Syntetos, A. A. and J. E. Boylan (2005). The accuracy of intermittent demand estimates. International Journal of Forecasting 21(2): 303-314. Syntetos, A. A., J. E. Boylan and J. D. Croston (2005). On the categorisation of demand patterns. Journal of the Operational Research Society 56(5): 495-503. M.Mohammadipour, 2009, References 270 Teles, P. and W. W. S. Wei (2002). The use of aggregate time series in testing for Gaussianity. Journal of Time Series Analysis 23(1): 95-116. Teunter, R. and B. Sani (2009). On the bias of Croston?s forecasting method. European Journal of Operational Research 194(1): 177-183. Teunter, R. H. and L. Duncan (2009). Forecasting intermittent demand: a comparative study. Journal of the Operational Research Society 60(3): 321-329. Tiao, G. C. and D. Xu (1993). Robustness of maximum likelihood estimates for multi-step predictors: the exponential smoothing case. Biometrika 80(3): 623-641. Venkataraman, K. N. (1982). A time series approach to the study of the simple subcritical Galton-Watson process with immigration. Advances in Applied Probability 14(1): 1-20. Wald, A. and J. Wolfowitz (1940). On a test whether two samples are from the same population. Annals of Mathematical Statistics 11(2): 147-162. Wei?, C. H. (2008a). The combined INAR(p) models for time series of counts. Statistics & Probability Letters 78(13): 1817-1822 Wei?, C. H. (2008b). Serial dependence and regression of Poisson INARMA models. Journal of Statistical Planning and Inference 138(10): 2975-2990. Wei?, C. H. (2008c). Thinning operations for modeling time series of counts-a survey. AStA Advances in Statistical Analysis 92(3): 319-341. Willemain, T. R., C. N. Smart and H. F. Schwarz (2004). A new approach to forecasting intermittent demand for service parts inventories. International Journal of Forecasting 20(3): 375-387. Willemain, T. R., C. N. Smart, J. H. Shocker and P. A. DeSautels (1994). Forecasting intermittent demand in manufacturing: a comparative evaluation of Croston?s method. International Journal of Forecasting 10(4): 529?538. Williams, T. (1984). Stock control with sporadic and slow-moving demand. Journal of the Operational Research Society 35(10): 939-948. Zeger, S. (1988). A regression model for time series of counts. Biometrika 75(4): 621-629. Zeger, S. L. and B. Qaqish (1988). Markov regression models for time series: a quasilikelihood approach. Biometrics 44(4): 1019-1031. Zheng, H., I. V. Basawa and S. Datta (2006). Inference for pth-order random coefficient integer-valued autoregressive processes. Journal of Time Series Analysis 27(3): 411-440. M.Mohammadipour, 2009, References 271 Zheng, H., I. V. Basawa and S. Datta (2007). First-order random coefficient integer- valued autoregressive processes. Journal of Statistical Planning and Inference 137(1): 212-229 Zhu, R. and H. Joe (2006). Modelling count data time series with Markov processes based on Binomial thinning. Journal of Time Series Analysis 27(5): 725?738. M.Mohammadipour, 2009, Appendix 3.A 272 Appendix 3.A Autocorrelation Function of an INARMA(1,1) Model In this appendix, we show how to obtain the ACF of an INARMA(1,1) process of ??=?????1 +??+?????1. The unconditional variance of the INARMA(1,1) process can be obtained from: var????= var??????1?+ var????+ var??????1?+ 2cov(?????1,?????1) Considering the fact that cov????,????=??cov(?,?), the above equation can be written as: var????=? 2var????1?+??1????????1?+?? 2 +?2?? 2 +??1????? +2cov???2????2 +?????1 +??????2?,?????1? =?2var????1?+??1??? ??(1 +?) 1?? +?? 2 +?2?? 2 +??1????? + 2???? 2 Hence, the unconditional variance of an INARMA(1,1) process is: var????= 1 1??2 [??+?+????2???+ (1 +? 2 + 2??)?? 2] Equation 3.A-1 If a Poisson distribution is assumed for innovations (??=?? 2 =?), the above result can be simplified to: var????= ? 1??2 [1 +?+?+ 3??] Equation 3.A-2 The INARMA(1,1) process, ??, can be written in terms of ???? as follows: ??=? ??????+? ? ?????? ??1 ?=0 +? ?????????1 ??1 ?=0 Equation 3.A-3 The autocovariance at lag ? is defined as: ??= cov(??,????) Equation 3.A-4 M.Mohammadipour, 2009, Appendix 3.A 273 Therefore, the autocovariance at lag ? is: ??= cov?????,?? ??????+? ? ?????? ??1 ?=0 +? ?????????1 ??1 ?=0 ?? =??var??????+ cov?????,? ? ?????? ??1 ?=0 ?+ cov?????,? ? ????????1 ??1 ?=0 ? =??var??????+ cov?????,? ??1??????? =??var??????+ cov????????1 +????+???????1,? ??1??????? By substitution from Equation 3.A-1: ??= ?? 1??2 ???+?+????2???+?1 +? 2 + 2????? 2?+???1??? 2 =???1?? ?(?+?+????2) 1??2 ???+? ?(1 +?2 + 2??) 1??2 +???? 2? As a result: ??=? ??1?? ?2 +??+?2????2 1??2 ???+? ?+??2 +?2?+? 1??2 ??? 2? Equation 3.A-5 Consequently, the ACF of an INARMA(1,1) is: ??=? ??2 +??+?2????2???+ (?+?? 2 +?2?+?)?? 2 ??+?+????2???+ (1 +?2 + 2??)?? 2 for ?= 1 ????1 for ?> 1 ? Equation 3.A-6 For a PoINARMA(1,1) with ??=?? 2 =?, the ACF would be: ??=? ?+?+??+?2 + 2?2? 1 +?+?+ 3?? for ?= 1 ????1 for ?> 1 ? Equation 3.A-7 M.Mohammadipour, 2009, Appendix 3.B 274 Appendix 3.B The Unconditional Variance of an INARMA(p,q) Model The unconditional variance of an INARMA(p,q) process of ??=? ??????? ? ?=1 + ??+? ??????? ? ?=1 , can be written as follows, where all the covariance terms, to be found later, have been summarized using the expression ??cov? terms?: var????= var??1????1?+?+ var?????????+ var???? +var??1????1?+?+ var?????????+?cov? terms Equation 3.B-1 var????=?1 2var????1?+?1?1??1??????1?+?+?? 2var?????? +???1???????????+?? 2 +?1 2?? 2 +?1?1??1???+?+?? 2?? 2 +???1??????+?cov? terms Equation 3.B-2 Hence, assuming stationarity of the process, we have: var????= 1 1?? ?? 2? ?=1 ?? 1 +? ?? ? ?=1 1?? ?? ? ?=1 ? ??1??? ? ?=1 +? ??1??? ? ?=1 ???? ?+?1 +? ?? 2 ? ?=1 ??? 2 +?cov? terms? Equation 3.B-3 Next, we focus on finding the covariance terms. There are two types of covariance: 1. The covariance between {??}s at different lags: cov(??,??) for ???. 2. The covariance between {??}s and {??}s: cov(??,??) for ???. For ?>?, the random disturbance terms are independent of previous observations. M.Mohammadipour, 2009, Appendix 3.B 275 The first group of covariance terms is given by: first Cov = 2cov??1????1,?2????2?+ 2cov??1????1,?3????3?+? +2cov??1????1,???????? +2cov??2????2,?3????3?+ 2cov??2????2,?4????4?+? + 2cov??2????2,????????+? + 2cov????2?????+2,???1?????+1?+ 2cov????2?????+2,???????? + 2cov????1?????+1,???????? first Cov = 2?1?2?1 + 2?1?3?2 +?+ 2?1?????1 +2?2?3?1 + 2?2?4?2 +?+ 2?2?????2 +? +2 ???2???1?1 + 2???2???2 +2???1???1 first Cov = 2? ???+1?1 ??1 ?=1 + 2? ???+2?2 ??2 ?=1 +?+ 2 ? ???+???1???1 ?????1? ?=1 first Cov = 2? ? ???+??? ??? ?=1 ??1 ?=1 Equation 3.B-4 Now we focus on the second type of covariance which is cov(??,??) for ???. Here there are three cases: 1. ?? Each of these cases is discussed here separately. M.Mohammadipour, 2009, Appendix 3.B 276 1. cov(??,??) when ?? Figure 3.B-3 in an INARMA(p,q) process when Second Cov 3 = 2cov??1????1,?1????1?+ 2cov??1????1,?2????2?+? +2cov??1????1,???????? +2cov??2????2,?2????2?+ 2cov??2????2,?3????3?+? + 2cov??2????2,????????+? +2cov????1?????+1,???1?????+1? + 2cov????1?????+1,???????? +2cov????????,???????? Then, the first terms in each of the rows are summed vertically, with the remaining terms in the rows being summed horizontally (see Figure 3.B-3). Second Cov 3 = 2?? 2? ??? ? ?=1 + 2? ?1?????1 ?? ? ?=2 + 2? ?2?????2 ?? ? ?=3 +? +2? ???1?????(??1) ?? ? ?=? ),cov( ji ZY qp ? ???1 ???1 ???1 ???2 ? ???1 ????+1 ???1 ???? ???2 ???2 ? ???2 ????+1 ???2 ???? ???? ???? ? ? M.Mohammadipour, 2009, Appendix 3.B 279 Second Cov 3 = 2?? 2? ??? ? ?=1 + 2? ? ??????? ?? ? ?=?+1 ??1 ?=1 Equation 3.B-7 The Equation 3.B-7 can be written as 2?? 2? ??? ? ?=1 + 2? ? ??????? ??? ?=?+1 ? ?=1 for generalization purposes because for ?=? the second summation would be (? )??=?+1 which has zero terms. Therefore, based on the results of Equation 3.B-5, Equation 3.B-6 and Equation 3.B- 7, the second group of covariance is given by: Second Cov = 2 ? ??? min (?,?) ?=1 ?? 2 + 2 ? ? ??????? ?? ? ?=?+1 min (?,?) ?=1 Equation 3.B-8 Finally, from the Equation 3.B-3, Equation 3.B-4, and Equation 3.B-8, the unconditional variance of an INARMA(p,q) process is: var????= 1 1?? ?? 2? ?=1 ?? 1 +? ?? ? ?=1 1?? ?? ? ?=1 ? ??1??? ? ?=1 +? ??1??? ? ?=1 ???? +?1 +? ?? 2 ? ?=1 ??? 2 + 2? ? ???+??? ??? ?=1 ??1 ?=1 ?+2 ? ??? min (?,?) ?=1 ?? 2 + 2 ? ? ??????? ?? ? ?=?+1 min (?,?) ?=1 var????= ?? 1?? ?? 2? ?=1 ? 1 +? ?? ? ?=1 1?? ?? ? ?=1 ? ??1??? ? ?=1 +? ??1??? ? ?=1 ? + ?? 2 1?? ?? 2? ?=1 ?1 +? ?? 2 ? ?=1 + 2 ? ??? min (?,?) ?=1 + 2? ? ???+??? ??? ?=1 ??1 ?=1 + 2? ? ??????? ??? ?=?+1 min (?,?) ?=1 1?? ?? 2? ?=1 Equation 3.B-9 M.Mohammadipour, 2009, Appendix 3.C 280 Appendix 3.C The Cross-Covariance Function between Y and Z for an INARMA(p,q) Model The cross-covariance function, ?? ??, is the covariance between ? and ? at lag ? and is defined by ?? ??=?(??????). Therefore, the cross-covariance at lag zero is given by: ?0 ??= cov???,???= cov?? ??????? ? ?=1 +??+? ??????? ? ?=1 ,??? Considering the fact that the innovation terms are independent of previous observations, we have: ?0 ??= cov???,???=???????=?? 2 The cross-covariance at lag one can be obtained from: ?1 ??= cov???,???1? = cov??1????1 +?+???????+??+?1????1 +?+???????,???1? = cov??1????1,???1?+ cov??1????1,???1?=?1?? 2 +?1?? 2 The cross-covariance at lag ? (0????) is given by: ?? ??= cov???,????? = cov??1????1 +?+???????+??+?1????1 +?+???????,????? = cov??1????1,?????+?+ cov????1???????1?,????? +cov????????,?????+???? 2 =???? 2 +?1???1 ?? +?+???1?1 ??+???? 2 ?? ??=???? 2 +? ????? ?? ? ?=1 Equation 3.C-1 M.Mohammadipour, 2009, Appendix 3.D 281 Appendix 3.D Over Lead Time Aggregation of an INAR(1) Model In this appendix, we show how to derive the conditional first and second moments of a lead time aggregated PoINAR(1) process (??=?? 2 =?). The aggregated process over lead time can be written as: ? ??+? ?+1 ?=1 =?????+??+1?+?????+1 +??+2?+?+?????+?+??+?+1? =?????+??+1?+????????+??+1?+??+2?+?+?????+?+??+?+1? =?????+??+1?+?? 2???+????+1 +??+2?+? +???+1???+? ????+1 +? ??1???+2 +?+????+?+??+?+1? It can be simplified as ? ??+? ?+1 ?=1 =?????+? 2???+?+? ?+1???? +???+1 +????+1 +?+? ????+1? +???+2 +????+2 +?+? ??1???+2?+?+???+?+????+? +??+?+1 Equation 3.D-1 The conditional expected value of the aggregated process is given by: ??? ??+? ?+1 ?=1 |???=????+? 2??+?+? ?+1???+??+??+?+? ??? +??+??+?+???1??+?+??+???+? ??? ??+? ?+1 ?=1 |???= ?(1???+1) 1?? ??+ ? 1?? ??1???+1?+?1??? +?+?1???? Therefore, we have: ??? ??+? ?+1 ?=1 |???= ?(1???+1) 1?? ??+ ? 1?? ???+ 1??? ?? ?+1 ?=1 ? Equation 3.D-2 Now we want to find the conditional variance of the aggregated process. We know that: M.Mohammadipour, 2009, Appendix 3.D 282 cov?????,?????=???????2??????????????=????var(?) Equation 3.D-3 Hence, we have cov??????,? ?????=? ????. The variance of the Equation 3.D-1 given ?? is: var?? ??+? ?+1 ?=1 |???= var??????+ var?? 2????+?+ var?? ?+1???? +2cov?????,? 2????+ 2cov?????,? 3????+?+ 2cov?????,? ?+1???? + 2cov??2???,? 3????+ 2cov?? 2???,? 4????+?+ 2cov?? 2???,? ?+1???? +?+ 2cov????1???,? ?????+ 2cov?? ??1???,? ?+1???? +2cov??????,? ?+1???? +var???+1?+ var?????+1?+ var?? 2???+1?+?+ var?? ????+1? +2cov???+1,????+1?+ 2cov???+1 ,? 2???+1?+?+ 2cov???+1 ,? ????+1? +2cov?????+1 ,? 2???+1?+ 2cov?????+1 ,? 3???+1?+? + 2cov?????+1 ,? ????+1? +?+ 2cov????2???+1 ,? ??1???+1?+ 2cov?? ??2???+1 ,? ????+1? +2cov????1???+1 ,? ????+1? +var???+2?+ var?????+2?+?+ var?? ??1???+2? +2cov???+2,????+2?+ 2cov???+2 ,? 2???+2?+?+ 2cov???+2 ,? ??1???+2? +2cov?????+2 ,? 2???+2?+ 2cov?????+2 ,? 3???+2?+? + 2cov?????+2 ,? ??1???+2? +?+ 2cov????3???+2 ,? ??2???+2?+ 2cov?? ??3???+2 ,? ??1???+2? +2cov????2???+2 ,? ??1???+2?+? +var???+??1?+ var?????+??1?+ var?? 2???+??1? +2cov???+??1 ,????+??1?+ 2cov???+??1 ,? 2???+??1? + 2cov?????+??1 ,? 2???+??1? +var???+?+ var?????+? +2cov???+?,????+? +var???+?+1? Since ?? is fixed, cov?? ????,? ?????=? ???var????= 0, using Equation 3.D-3. M.Mohammadipour, 2009, Appendix 3.D 283 var?? ??+? ?+1 ?=1 |???=?(1??)?????+? 2(1??2)?????+?+? ?+1(1???+1)????? +?+??2?+??1?????+??4?+?2?1??2???+?+??2??+???1?????? +2??+?2 +?+???+ 2??3 +?4 +?+??+1??+?+ 2??2??3 +?2??2?? + 2??2??1?? +?+??2?+??1?????+??4?+?2?1??2???+? +??2??2?+???1?1????1??? +2??+?2 +?+???1??+ 2??3 +?4 +?+???+?+ 2??2??5 +?2??4?? + 2??2??3??+? +?+??2?+??1?????+??4?+?2?1??2??? +2??+?2??+ 2[?3]? +?+??2?+??1????? +2?? +? The above result can be summarized to: var?? ??+? ?+1 ?=1 |???=?(1??)??+? 2(1??2)??+?+? ?+1(1???+1)?? +?+????+??2??+?+????? +2??+?2 +?+???+ 2??3 +?4 +?+??+1??+?+ 2??2??3 +?2??2?? + 2??2??1?? +?+????+??2??+?+????1?? +2??+?2 +?+???1??+ 2??3 +?4 +?+???+?+ 2??2??5 +?2??4?? + 2??2??3??+? +?+????+??2?? +2??+?2??+ 2[?3]? +?+???? +2?? +? var?? ??+? ?+1 ?=1 |???=?(1??)??+? 2(1??2)??+?+? ?+1(1???+1)?? +??1 +?+?+???+??1 +?+?+???1?+?+??1 +?+?2?+??1 +?? +? +2??+?2 +?+???+ 2??3 +?4 +?+??+1??+?+ 2??2??3 +?2??2?? + 2??2??1?? +2??+?2 +?+???1??+ 2??3 +?4 +?+???+?+ 2??2??5 +?2??4?? + 2??2??3??+? +2??+?2??+ 2[?3]? +2?? M.Mohammadipour, 2009, Appendix 3.D 284 var?? ??+? ?+1 ?=1 |???=??? ? ??1???? ?+1 ?=1 + ? 1?? ???+ 1??? ?? ?+1 ?=1 ? +2???1 +?+?+??1?+ 2?3??1 +?+?+??2?+? +2?2??3??1 +??+ 2?2??1? +2???1 +?+?+??2?+ 2?3??1 +?+?+??3?+? +2?2??5??1 +??+ 2?2??3?+? +2???1 +??+ 2?3? +2?? So var?? ??+? ?+1 ?=1 |???=??? ? ??1???? ?+1 ?=1 + ? 1?? ???+ 1??? ?? ?+1 ?=1 ? + 2?? 1?? ?1????+ 2?3? 1?? ?1???1?+?+ 2?2??3? 1?? ?1??2?+ 2?2??1? 1?? [1??] + 2?? 1?? ?1???1?+ 2?3? 1?? ?1???2?+?+ 2?2??5? 1?? ?1??2? + 2?2??3? 1?? ?1??? +? + 2?? 1?? ?1??2?+ 2?3? 1?? ?1??? + 2?? 1?? ?1??? Summing the above expressions vertically results in: var?? ??+? ?+1 ?=1 |???=??? ? ??1???? ?+1 ?=1 + ? 1?? ???+ 1??? ?? ?+1 ?=1 ? + 2?? 1?? ???(?+?2 +?+??)?+ 2?3? 1?? ????1??(?+?2 +?+??1)? +?+ 2?2??3? 1?? ?2?(?+?2)?+ 2?2??1? 1?? [1??] Finally, the conditional variance of the aggregated process is: M.Mohammadipour, 2009, Appendix 3.D 285 var?? ??+? ?+1 ?=1 |???=??? ? ??1???? ?+1 ?=1 + ? 1?? ???+ 1??? ?? ?+1 ?=1 ? + 2? 1?? ? ?2??1?????+ 1?? ?(1?????+1) 1?? ? ? ?=1 Equation 3.D-4 M.Mohammadipour, 2009, Appendix 4.A 286 Appendix 4.A Infinite Autoregressive Representation of an INARMA(p,q) Model It will be shown in this appendix that the Infinite Auto-Regressive Representation (IARR) of an INARMA(p,q) process (??=? ??????? ? ?=1 +??+? ??????? ? ?=1 ) is as follows: ??=??+? ? ??????? ?? ?=1 ? ?=1 Equation 4.A-1 where ??= ? ? ? ? ? ?? ???? ? ?=1 + 1 0 ? ? Equation 4.A-2 Then ?? can be expressed in terms of {????}?=0 ? as: ??=???? ? ??????? ?? ?=1 ? ?=1 Equation 4.A-3 where ?? is given by Equation 4.A-2. When expressing the INARMA(p,q) process in terms of {????}?=1 ? , it should be noted that because ???+????(?+?)??, these coefficients cannot be added. Therefore, another summation over the number of ???? terms has been used (? ) ?? ?=1 . First, it is shown that the expression for ?? is correct when ?>?. An example will motivate the general case. Consider an INARMA(2,1) process of: ??=?1????1 +?2????2 +??+?1????1 M.Mohammadipour, 2009, Appendix 4.A 287 We are interested to find the number of ???3 terms in the IARR of this process. It can be seen that: ???1 =???1??1????2??2????3??1????2 ???2 =???2??1????3??2????4??1????3 ???3 =???3??1????4??2????5??1????4 The INARMA(2,1) process can be written as: ??=?1????1 +?2????2 +??+?1?????1??1????2??2????3??1????2? =?1????1 +?2????2 +??+?1????1??1?1????2??2?1????3??1 2 ?(???2??1????3??2????4??1????3) =?1????1 +?2????2 +??+?1????1??1?1????2??2?1????3??1 2????2 +?1?1 2????3 +?2?1 2????4 +?1 3?(???3??1????4??2????5??1????4) It can be seen that ???3 terms come from ???1, ???2, and ???3 terms. It can be easily shown that ???4 terms come from ???2, ???3, and ???4 terms. Using the same argument, it can be shown that when ?>?, ???? terms come from {????+?}?=0 ? terms, with each of them producing one ????. Now, we want to check if, for ?>?, ?? is in fact ? ???? ? ?=1 (??=???1 +???2 +?+ ????). We know that ?? is the number of ???? in IARR of an INARMA(p,q) process, so: ???1 = the number of ????+1 ???2 = the number of ????+2 ? ????= the number of ????+? which come from which come from which come from ? ? ? ? {????+1+?}?=0 ? {????+2+?}?=0 ? ? {????+?+?}?=0 ? The above terms can be written as follows: M.Mohammadipour, 2009, Appendix 4.A 288 ????+1 ????+2 ? ????+? ????+2 ????+3 ? ????+?+1 ????+3 ????+4 ? ????+?+2 ? ? ? ????+?+1 ????+?+2 ? ????+?+? If we look at the elements that produce ???? (which is the set {????+?}?=0 ? = {????,????+1,?,????+?}) we can see that: 1. the number of ???? comes from {????+1,????+2,?,????+?} and as it can be seen in the above table, these are shown by the first rectangle. 2. the number of ????+1 comes from {????+2,????+3,?,????+?+1} and as it can be seen in the above table, these are shown by the second rectangle. 3. the number of ????+2 comes from {????+3,????+4,?,????+?+2} and as it can be seen in the above table, these are shown by the third rectangle. 4. the number of ????+? comes from {????+?+1,????+?+2,?,????+?+?} and as it can be seen in the above table, these are shown by the last rectangle. Therefore, ??=???1 +???2 +?+????. For the case of 0 2. As previously mentioned, when ??2, the number of ?? in ??+? is equal to the number of ?? in the two previous terms plus one. The corresponding coefficient for ?? in each of {??+?}?=1 ?+1 (say ??+3) is obtained from ?1 thinned the coefficient of ?? in the previous term (in this case ??+2) and ?2 thinned the coefficient of ?? in the next previous term (in this case ??+1). These coefficients are shown in Table 6.A-1. Table 6.A-1 Coefficients of in each of for an INAR(2) model ?= 1,?= 1 ?11 1 =?1 ?= 2,?= 1,2 ?12 1 =?1 2 ?22 1 =?2 ? ? ?=?+ 1,?= 1,?,??+1 1 ?1(?+1) 1 =?1 ?+1 ?2(?+1) 1 =?1 ??1?2 ? The number of ???1 in each of {??+?}?=1 ?+1 is also equal to the sum of the number of ?? in the two previous terms (??+??1,??+??2). Therefore, ?? 2 =???1 2 +???2 2 for ?> 1. But, for ??1 this is equal to the number of ???1 in the two previous terms plus one. tY 11 ? ?? l jjtY }{ M.Mohammadipour, 2009, Appendix 6.A 295 The corresponding coefficient for ???1 in each of {??+?}?=1 ?+1 (say ??+3) is obtained from ?1 thinned the coefficient of ???1 in the previous term (in this case ??+2) and ?2 thinned the coefficient of ???1 in the next previous term (in this case ??+1). These coefficients are shown in Table 6.A-2. Table 6.A-2 Coefficients of in each of for an INAR(2) model ?= 1,?= 1 ?11 2 =?2 ?= 2,?= 1 ?12 2 =?1?2 ?= 3,?= 1,2 ?13 2 =?1 2?2 ?23 2 =?1?2 ? ? ?=?+ 1,?= 1,?,??+1 2 ?1(?+1) 2 =?1 ??2 ?2(?+1) 2 =?1 ??1?2 ? It can be seen from Equation 6.A-2 that because the process is an autoregressive process of order two, the number of ??+?? increases in each of {??+?}?=1 ?+1 . For the INAR(2) case, this number, shown by ?? 3, can be obtained from the number of ??+?? in the two previous terms plus one because each of {??+?}?=1 ?+1 has one ??+? as well. The corresponding coefficient for each ??+??, shown by ?? 3 , is ?1 thinned the coefficients of ??+?? in the previous term (??+??1) and ?2 thinned the coefficients of ??+?? in the next previous term (??+??2). The coefficient for ??+? in each of {??+?}?=1 ?+1 is one. ?+?? is the subscripts of innovation terms in each of {??+?}?=1 ?+1 which from Equation 6.A-2 it can be seen that ?? is given by: ??=? ??(??2) for 1??????2 3 ??(??1) for ???2 3 2, ?? equals to all the subscripts included in the two 1?tY 11 ? ?? l jjtY }{ M.Mohammadipour, 2009, Appendix 6.A 296 previous {??+?}?=1 ?+1 plus ?. This means that ?? is as shown in Equation 6.A-4. The corresponding coefficient for each ??+?? and the subscript of innovation terms, ?+?? are shown in Table 6.A-3. Based on Equation 6.A-3, the conditional expected value of the aggregated process is: ??? ??+? ?+1 ?=1 |???=?? ? ?? 1 ?? 1 ?=1 ?+1 ?=1 ???+?? ? ?? 2 ?? 2 ?=1 ?+1 ?=1 ????1 +?? ? ?? 3 ?? 3 ?=1 ?+1 ?=1 ?? Equation 6.A-5 Table 6.A-3 Coefficients of in each of for an INAR(2) model ?= 1 ?= 1,?,?1 3 where ?1 3 = 1 ?11 3 = 1 ?11 = 1 ?= 2 ?= 1,?,?2 3 where ?2 3 = 2 ?12 3 =?1 ?22 3 = 1 ?12 = 1 ?22 = 2 ?= 3 ?= 1,?,?3 3 where ?3 3 = 4 ?13 3 =?1 2 ?23 3 =?1 ?33 3 =?2 ?43 3 = 1 ?13 = 1 ?23 = 2 ?33 = 1 ?43 = 3 ? ? ? ?=?+ 1 ?= 1,?,??+1 3 ?1(?+1) 3 =?1 ? ?2(?+1) 3 =?1 ??1 ? ?1(?+1) = 1 ?2??+1?= 2 ? ijkt Z ? 1 1 ? ?? l jjtY }{ M.Mohammadipour, 2009, Appendix 6.B 297 Appendix 6.B Lead Time Forecasting for an INARMA(1,2) Model For the INARMA(1,2) process of ??=?????1 +??+?1????1 +?2????2, the cumulative ? over lead time ? is given by: ? ??+? ?+1 ?=1 =??+1 +??+2 +?+??+?+1 =?????+??+1 +?1???+?2????1? +?????+1 +??+2 +?1???+1 +?2????+?+ +?????+?+??+?+1 +?1???+?+?2???+?1? Equation 6.B-1 In order to find the conditional expectation of the aggregated process, Equation 6.B-1 should be expressed in terms of ??. ? ??+? ?+1 ?=1 =?????+??+1 +?1???+?2????1? +??2???+????+1 +??1???+??2????1 +??+2 +?1???+1 +?2???? +??3???+? 2???+1 +? 2?1???+? 2?2????1 +????+2 +??1???+1 +??2???+??+3 +?1???+2 +?2???+1?+? Equation 6.B-2 The above expression is not an infinite series but a finite series where the remaining terms can be obtained by repeated substitution. The above equation can be written as: ? ??+? ?+1 ?=1 =? ? ?? 1 ??? ?? 1 ?=1 ?+1 ?=1 +? ? ?? 2 ???+?? ?? 2 ?=1 ?+1 ?=1 Equation 6.B-3 where ?? 1 is the number of ?? terms in each of {??+?}?=1 ?+1 and ?? 1 is the corresponding coefficient for each ??. ?? 2 is the number of ??+?? terms in each of {??+?}?=1 ?+1 , ?? 2 is the corresponding coefficient for each ??+??. Each of these terms is explained below. It can be seen from Equation 6.B-2 that because the process is an integer autoregressive of order one, each of {??+?}?=1 ?+1 only has one ?? and therefore ?? 1 = 1 (because the number of ?? in each of {??+?}?=1 ?+1 is equal to the number of ?? in the previous term (??+??1)). Therefore, the corresponding coefficient for ?? in each of M.Mohammadipour, 2009, Appendix 6.B 298 {??+?}?=1 ?+1 (say ??+2) is obtained from ? thinned the coefficient of ?? in the previous term (in this case ??+1). Therefore, ?? 1 =??. These coefficients are shown in Table 6.B-1. Table 6.B-1 Coefficients of in each of for an INARMA(1,2) model ?= 1,?= 1 ?11 1 =? ?= 2,?= 1 ?12 1 =?2 ? ? ?=?+ 1,?= 1 ?1(?+1) 1 =??+1 It can be seen from Equation 6.B-2 that because the process has an autoregressive component of order one and also a moving average component of order two, the number of ??+?? increases in each of {??+?}?=1 ?+1 . Each of {??+?}?=1 ?+1 has three ??+?? and also all the ??+?? terms of the previous ? element (??+??1). Therefore, this number, shown by ?? 2, can be obtained from ???1 2 + 3. The same argument applies for the INARMA(p,q) case where the number of ??+?? is equal to the number of them in the ? previous terms plus ?+ 1 (see section 6.3.4). ?+?? is the subscripts of innovation terms in each of {??+?}?=1 ?+1 which from Equation 6.B-2 it can be seen that ?? is given by: ??=? ??(??1) for ?= 1,?,???1 2 ?,??1,??2 for ?=???1 2 + 1,?,?? 2 ? for ?= 1,?,?+ 1 Equation 6.B-4 For example, for ?= 1, the subscript of innovation terms is ??1 =? 1 ?= 1 0 ?= 2 ?1 ?= 3 ?, because as can be seen from Equation 6.B-1, there are three innovation terms in the expression for ??+1 which are {??+1,??,???1}. For ?> 1, the ??+??s come from the innovation terms included in the previous {??+?}?=1 ?+1 plus {?,??1,??2}. This means that ?? is as shown in Equation 6.B-4. The corresponding coefficient for each ??+?? and the subscript of innovation terms, ?+?? are shown in Table 6.B-2. tY 11 ? ?? l jjtY }{ M.Mohammadipour, 2009, Appendix 6.B 299 Based on Equation 6.B-3, the conditional expected value of the aggregated process is: ??? ??+? ?+1 ?=1 |???=?? ? ?? 1 ?? 1 ?=1 ?+1 ?=1 ???+?? ? ?? 2 ?? 2 ?=1 ?+1 ?=1 ?? Equation 6.B-5 Table 6.B-2 Coefficients of in each of for an INARMA(1,2) model ?= 1 ?= 1,?,?1 2 where ?1 2 = 3 ?11 2 = 1 ?21 2 =?1 ?31 2 =?2 ?11 = 1 ?21 = 0 ?31 =?1 ?= 2 ?= 1,?,?2 2 where ?2 2 = 6 ?12 2 =? ?22 2 =??1 ?32 2 =??2 ?42 2 = 1 ?52 2 =?1 ?62 2 =?2 ?12 = 1 ?22 = 0 ?32 =?1 ?42 = 2 ?52 = 1 ?62 = 0 ?= 3 ?= 1,?,?3 2 where ?3 2 = 9 ?13 2 =?2 ?23 2 =?2?1 ?33 2 =?2?2 ?43 2 =? ?53 2 =??1 ?63 2 =??2 ?73 2 = 1 ?83 2 =?1 ?93 2 =?2 ?13 = 1 ?23 = 0 ?33 =?1 ?43 = 2 ?53 = 1 ?63 = 0 ?73 = 3 ?83 = 2 ?93 = 1 ? ? ? ijkt Z ? 1 1 ? ?? l jjtY }{ M.Mohammadipour, 2009, Appendix 6.C 300 Appendix 6.C Lead Time Forecasting for an INARMA(p,q) Model In order to find the conditional mean of the over-lead-time-aggregated process, we need to express the aggregated INARMA(p,q) process in terms of the last ? observations (????+1,????+2,?,???1,??). The aggregated process is given by: ? ??+? ?+1 ?=1 =??+1 +??+2 +?+??+?+1 Equation 6.C-1 Each of the {??+?}?=1 ?+1 in the RHS of the above equation needs to be expressed in terms of {????+1}?=1 ? by repeated substitution of ??+? in Equation 3-50. Because the autoregressive order of the process is ?, ??+? can be expressed in terms of ? previous observations by ?1???+??1 +?+?????+???. Now, if ????(??1), as mentioned in Appendix 6.A, there is one ???(??1) when we express the ?th observation in the RHS of the Equation 6.C-1 (??+?) without any need for repeated substitution. Repeated substitution of (??+1,?,??+?????1?) by their ? previous observations would result in obtaining more ???(??1). Therefore, in total, the number of ???(??1) in each of {??+?}?=1 ?+1 when ????(??1) is equal to the number of ???(??1) in its ? previous observations plus one. However, as explained in Appendix 6.A, when ?>??(??1), each ??+? from Equation 6.C-1 should be substituted by Equation 3-50 in order to reach ???(??1), and the number of ???(??1) in each of {??+?}?=1 ?+1 would be equal to the number of ???(??1) in its ? previous observations. For ????(??1), the corresponding coefficient of ???(??1) in the ?th observation in the RHS of the Equation 6.C-1 (??+?) is ??+(??1) because: ??+?=?1???+??1 +?+??+(??1)????(??1) +?+?????+???+??+?+? ?????+??? ? ?=1 For other ???(??1) the coefficient in each of {??+?}?=1 ?+1 is ?? thinned the coefficient of ???(??1) in the ?th previous observation for ?= 1,?,?. M.Mohammadipour, 2009, Appendix 6.C 301 For ?>??(??1), again, the coefficient of ???(??1) in each of {??+?}?=1 ?+1 is ?? thinned the coefficient of ???(??1) in the ?th previous observation for ?= 1,?,? (the difference with the previous case is that we do not have ??+(??1)). Now we come back to Equation 6.C-1 to find the ? terms in each of {??+?}?=1 ?+1 in the RHS of the equation when they expressed in terms of {????+1}?=1 ? . As the process has a moving average component of order ?, each {??+?}?=1 ?+1 has ?+ 1 innovation terms {??+?,??+??1,?,??+???}. However, as mentioned in Appendix 6.B, by repeated substitution each {??+?}?=1 ?+1 can be expressed in terms of ? previous observations, each also with ?+ 1 innovation terms. Therefore, the total number of innovation terms in each of {??+?}?=1 ?+1 is equal to the number of innovation terms in the ? previous observations, plus ?+ 1. The corresponding coefficients for the ?+ 1 terms {??+?,??+??1,?,??+???} are {1,?1,?,??}, respectively. For the innovation terms that come from the ? previous observations, coefficients would be ?? thinned the coefficient of ??+?? in the ?th previous observation for ?= 1,?,?. ?+?? denotes the subscript of ? for each ?,? (?= 1,?,?+ 1 and ?= 1,?,?? ?+1 ). As previously mentioned, each {??+?}?=1 ?+1 has ?+ 1 innovation terms {??+?,??+??1,?,??+???}. Therefore, the subscripts for the last ?+ 1 innovation terms in each {??+?}?=1 ?+1 are {???,??1,?,?}. This is shown in Table 6.C-1 by ?=???1 ?+1 + 1,?,???1 ?+1 +?? ?+1 . The other subscripts of innovation terms in each of {??+?}?=1 ?+1 simply are the subscripts of the innovation terms of ? previous observations. As a result, the aggregated process can be expressed as Equation 6-26 with the associated parameters as defined in Table 6.C-1. M.Mohammadipour, 2009, Appendix 6.C 302 Table 6.C-1 Parameters of the over-lead-time-aggregated INARMA(p,q) model fo r ? = 1 ,? ,? ?? ?=? ?? ???? ?? ?=1 + 1 ????(??1) ? ???? ?? ?=1 ?>??(??1) ? ?? ?= ? ? ? ? ? ? ? ? ? ? ? ? ????(???) ? ?= 1,?,???? ? ? ? ?1??(??1) ? ?=???2 ? + 1,?,???2 ? +???1 ? ??+(??1) ?=???1 ? + 1 ?????(??1) ? ????(???) ? ?= 1,?,???? ? ? ? ?1??(??1) ? ?=???2 ? + 1,?,???2 ? +???1 ? ??>??(??1) ? ?? ?+1 =?? ???? ?+1? ?=1 ?+ (?+ 1) ?? ?+1 = ? ? ? ? ? ????(???) ?+1 ?= 1,?,???? ?+1 ? ? ?1??(??1) ?+1 ?=???2 ?+1 + 1,?,???2 ?+1 +???1 ?+1 ??,?,?1 , 1 ?=???1 ?+1 + 1,?,???1 ?+1 +?? ?+1 ? ??= ? ? ? ? ? {??(???)} ?= 1,?,???? ?+1 ? ? {??(??1)} ?=? ???? ?+1? ?=2 + 1,?, (? ???? ?+1? ?=2 ) +???1 ?+1 ???,?,??1,? ?=? ???? ?+1? ?=1 + 1,?,?? ?+1 ? M.Mohammadipour, 2009, Appendix 8.A 303 Appendix 8.A The MSE of YW and CLS Estimates for INAR(1), INMA(1) and INARMA(1,1) Processes In this appendix, the two estimation methods used in this research are compared using MSE. Earlier in chapter 8, these methods have been compared in terms of their impact on forecast accuracy. The results for ?= 24, 36, 48, 96, 500 are presented as follows. Table 8.A-1 MSE of YW and CLS estimates of for an INAR(1) process Parameters YW CLS ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.0184 0.0167 0.0144 0.0097 0.0031 0.0219 0.0186 0.0154 0.0100 0.0031 ?= 0.5,?= 0.5 0.0742 0.0495 0.0359 0.0165 0.0030 0.0721 0.0481 0.0352 0.0161 0.0030 ?= 0.9,?= 0.5 0.1460 0.0700 0.0435 0.0129 0.0009 0.1120 0.0519 0.0333 0.0104 0.0008 ?= 0.1,?= 1 0.0196 0.0154 0.0138 0.0094 0.0026 0.0230 0.0167 0.0147 0.0097 0.0026 ?= 0.5,?= 1 0.0725 0.0450 0.0333 0.0144 0.0028 0.0707 0.0440 0.0324 0.0140 0.0028 ?= 0.9,?= 1 0.1546 0.0756 0.0456 0.0134 0.0008 0.1145 0.0582 0.0350 0.0109 0.0007 ?= 0.1,?= 3 0.0192 0.0152 0.0136 0.0084 0.0026 0.0223 0.0167 0.0145 0.0087 0.0026 ?= 0.5,?= 3 0.0679 0.0445 0.0319 0.0137 0.0021 0.0658 0.0430 0.0312 0.0135 0.0021 ?= 0.9,?= 3 0.1468 0.0725 0.0442 0.0119 0.0008 0.1105 0.0555 0.0339 0.0095 0.0007 ?= 0.1,?= 5 0.0192 0.0155 0.0129 0.0088 0.0027 0.0222 0.0170 0.0137 0.0091 0.0027 ?= 0.5,?= 5 0.0709 0.0446 0.0320 0.0142 0.0022 0.0689 0.0429 0.0313 0.0139 0.0022 ?= 0.9,?= 5 0.1455 0.0712 0.0437 0.0125 0.0008 0.1095 0.0550 0.0333 0.0101 0.0007 Table 8.A-2 Comparison of YW and CLS estimates of for an INAR(1) process Parameters YW CLS ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.0375 0.0243 0.0179 0.0100 0.0023 0.0380 0.0245 0.0181 0.0100 0.0023 ?= 0.5,?= 0.5 0.1072 0.0712 0.0473 0.0224 0.0034 0.1030 0.0692 0.0464 0.0220 0.0034 ?= 0.9,?= 0.5 3.9136 1.8446 1.1551 0.3370 0.0224 2.9900 1.3812 0.8930 0.2709 0.0206 ?= 0.1,?= 1 0.0867 0.0608 0.0452 0.0272 0.0060 0.0898 0.0625 0.0463 0.0275 0.0060 ?= 0.5,?= 1 0.3934 0.2350 0.1580 0.0713 0.0118 0.3852 0.2271 0.1545 0.0697 0.0118 ?= 0.9,?= 1 16.5122 7.6779 4.7735 1.3540 0.0790 12.1860 5.8605 3.6707 1.1016 0.0721 ?= 0.1,?= 3 0.3945 0.3001 0.2438 0.1413 0.0370 0.4275 0.3185 0.2528 0.1441 0.0372 ?= 0.5,?= 3 2.8535 1.6850 1.2493 0.5244 0.0824 2.7658 1.6303 1.2121 0.5149 0.0822 ?= 0.9,?= 3 132.6483 66.7297 40.2510 10.8786 0.6995 99.9406 51.2825 30.9385 8.7135 0.6386 ?= 0.1,?= 5 0.8928 0.6958 0.5521 0.3577 0.0968 0.9817 0.7426 0.5772 0.3659 0.0972 ?= 0.5,?= 5 7.4580 4.6569 3.4168 1.4914 0.2219 7.2026 4.4779 3.3367 1.4616 0.2204 ?= 0.9,?= 5 368.6249 180.0181 111.2582 31.8673 2.0634 278.5108 139.5061 84.8648 25.7175 1.8826 ? ? M.Mohammadipour, 2009, Appendix 8.A 304 Table 8.A-3 Comparison of YW and CLS estimates of for an INMA(1) process Parameters YW CLS ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.0516 0.0421 0.0349 0.0193 0.0039 0.0251 0.0216 0.0171 0.0097 0.0028 ?= 0.5,?= 0.5 0.1349 0.1008 0.0843 0.0493 0.0110 0.1147 0.0781 0.0721 0.0436 0.0211 ?= 0.9,?= 0.5 0.2161 0.1374 0.1024 0.0604 0.0121 0.2683 0.2008 0.1647 0.1131 0.0518 ?= 0.1,?= 1 0.0459 0.0364 0.0310 0.0166 0.0040 0.0236 0.0179 0.0152 0.0096 0.0029 ?= 0.5,?= 1 0.1233 0.0939 0.0822 0.0469 0.0106 0.1038 0.0805 0.0673 0.0417 0.0198 ?= 0.9,?= 1 0.2010 0.1485 0.1019 0.0548 0.0117 0.2752 0.2180 0.1726 0.1182 0.0517 ?= 0.1,?= 3 0.0468 0.0376 0.0297 0.0179 0.0038 0.0205 0.0172 0.0139 0.0096 0.0026 ?= 0.5,?= 3 0.1245 0.0960 0.0784 0.0474 0.0102 0.1110 0.0822 0.0696 0.0436 0.0207 ?= 0.9,?= 3 0.2141 0.1446 0.1156 0.0511 0.0113 0.3103 0.2391 0.2041 0.1321 0.0592 ?= 0.1,?= 5 0.0517 0.0400 0.0343 0.0154 0.0038 0.0214 0.0165 0.0151 0.0082 0.0027 ?= 0.5,?= 5 0.1222 0.0972 0.0829 0.0458 0.0095 0.1154 0.0902 0.0734 0.0427 0.0205 ?= 0.9,?= 5 0.2081 0.1388 0.0996 0.0563 0.0120 0.3434 0.2682 0.2226 0.1598 0.0695 Table 8.A-4 Comparison of YW and CLS estimates of for an INMA(1) process Parameters YW CLS ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.0364 0.0240 0.0194 0.0103 0.0021 0.0370 0.0238 0.0189 0.0098 0.0020 ?= 0.5,?= 0.5 0.0599 0.0419 0.0280 0.0137 0.0025 0.0665 0.0456 0.0310 0.0160 0.0041 ?= 0.9,?= 0.5 0.0739 0.0461 0.0305 0.0153 0.0025 0.0781 0.0541 0.0376 0.0193 0.0050 ?= 0.1,?= 1 0.0830 0.0576 0.0467 0.0240 0.0057 0.0834 0.0562 0.0444 0.0220 0.0052 ?= 0.5,?= 1 0.1637 0.1042 0.0851 0.0393 0.0074 0.1831 0.1152 0.0948 0.0448 0.0124 ?= 0.9,?= 1 0.2442 0.1441 0.0993 0.0397 0.0065 0.2823 0.1737 0.1229 0.0540 0.0162 ?= 0.1,?= 3 0.3812 0.2835 0.2296 0.1407 0.0371 0.3530 0.2541 0.1971 0.1166 0.0314 ?= 0.5,?= 3 1.0334 0.7039 0.5304 0.2635 0.0507 1.1603 0.7754 0.5886 0.3137 0.1004 ?= 0.9,?= 3 1.5474 0.9868 0.7012 0.2655 0.0430 1.9331 1.3063 0.9528 0.4364 0.1303 ?= 0.1,?= 5 0.8853 0.6958 0.5688 0.3405 0.0880 0.7923 0.5717 0.4578 0.2725 0.0737 ?= 0.5,?= 5 2.6253 1.8145 1.4464 0.6367 0.1212 3.0335 2.1586 1.6302 0.7749 0.2682 ?= 0.9,?= 5 4.1058 2.4764 1.5930 0.7347 0.1130 5.8606 3.9471 2.7201 1.4349 0.4352 ? ? M.Mohammadipour, 2009, Appendix 8.A 305 Table 8.A-5 Comparison of YW and CLS estimates of for an INARMA(1,1) process Parameters YW CLS ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.1,?= 0.5 0.2421 0.2299 0.2029 0.1588 0.0474 0.0498 0.0420 0.0334 0.0225 0.0136 ?= 0.1,?= 0.9,?= 0.5 0.0590 0.0423 0.0326 0.0225 0.0085 0.0685 0.0576 0.0484 0.0329 0.0087 ?= 0.5,?= 0.5,?= 0.5 0.1426 0.1129 0.0904 0.0458 0.0065 0.1035 0.0743 0.0586 0.0308 0.0056 ?= 0.9,?= 0.1,?= 0.5 0.2587 0.1328 0.0806 0.0197 0.0011 0.1228 0.0654 0.0425 0.0129 0.0009 ?= 0.1,?= 0.1,?= 1 0.2440 0.2409 0.1968 0.1470 0.0492 0.0490 0.0380 0.0299 0.0246 0.0136 ?= 0.1,?= 0.9,?= 1 0.0666 0.0381 0.0336 0.0212 0.0081 0.0808 0.0643 0.0546 0.0337 0.0093 ?= 0.5,?= 0.5,?= 1 0.1444 0.1108 0.0829 0.0412 0.0062 0.0850 0.0621 0.0478 0.0257 0.0051 ?= 0.9,?= 0.1,?= 1 0.2593 0.1266 0.0744 0.0196 0.0011 0.1093 0.0571 0.0372 0.0111 0.0008 ?= 0.1,?= 0.1,?= 5 0.2295 0.2187 0.2025 0.1606 0.0428 0.0434 0.0334 0.0275 0.0200 0.0127 ?= 0.1,?= 0.9,?= 5 0.0606 0.0418 0.0317 0.0219 0.0082 0.1121 0.0965 0.0858 0.0631 0.0194 ?= 0.5,?= 0.5,?= 5 0.1373 0.1032 0.0843 0.0392 0.0061 0.0618 0.0441 0.0327 0.0183 0.0058 ?= 0.9,?= 0.1,?= 5 0.2611 0.1321 0.0758 0.0184 0.0012 0.1033 0.0554 0.0343 0.0101 0.0008 Table 8.A-6 Comparison of YW and CLS estimates of for an INARMA(1,1) process Parameters YW CLS ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.1,?= 0.5 0.0470 0.0407 0.0353 0.0226 0.0109 0.0266 0.0205 0.0178 0.0118 0.0067 ?= 0.1,?= 0.9,?= 0.5 0.2646 0.1976 0.1597 0.0864 0.0158 0.4152 0.3694 0.3470 0.2715 0.1502 ?= 0.5,?= 0.5,?= 0.5 0.1520 0.1420 0.1405 0.1259 0.0495 0.1197 0.1128 0.1042 0.0948 0.0831 ?= 0.9,?= 0.1,?= 0.5 0.3237 0.2744 0.2342 0.1288 0.0189 0.0123 0.0100 0.0081 0.0063 0.0049 ?= 0.1,?= 0.1,?= 1 0.0492 0.0333 0.0305 0.0214 0.0106 0.0243 0.0181 0.0173 0.0119 0.0068 ?= 0.1,?= 0.9,?= 1 0.2876 0.2014 0.1632 0.0855 0.0163 0.4813 0.4255 0.3819 0.2880 0.1609 ?= 0.5,?= 0.5,?= 1 0.1488 0.1407 0.1386 0.1196 0.0460 0.1468 0.1280 0.1240 0.1146 0.0895 ?= 0.9,?= 0.1,?= 1 0.3443 0.2641 0.2224 0.1259 0.0164 0.0080 0.0075 0.0071 0.0063 0.0057 ?= 0.1,?= 0.1,?= 5 0.0455 0.0356 0.0296 0.0213 0.0118 0.0181 0.0152 0.0137 0.0108 0.0070 ?= 0.1,?= 0.9,?= 5 0.2685 0.2019 0.1575 0.0872 0.0149 0.6413 0.5912 0.5619 0.4657 0.2561 ?= 0.5,?= 0.5,?= 5 0.1492 0.1453 0.1421 0.1215 0.0483 0.1797 0.1752 0.1676 0.1621 0.1180 ?= 0.9,?= 0.1,?= 5 0.3464 0.2849 0.2332 0.1281 0.0184 0.0076 0.0075 0.0074 0.0074 0.0076 Table 8.A-7 Comparison of YW and CLS estimates of for an INARMA(1,1) process Parameters YW CLS ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.1,?= 0.5 0.1007 0.0847 0.0731 0.0534 0.0124 0.0486 0.0315 0.0234 0.0136 0.0033 ?= 0.1,?= 0.9,?= 0.5 0.0993 0.0634 0.0466 0.0213 0.0049 0.0953 0.0643 0.0509 0.0289 0.0151 ?= 0.5,?= 0.5,?= 0.5 0.1929 0.1195 0.0785 0.0312 0.0045 0.1869 0.1295 0.0942 0.0485 0.0179 ?= 0.9,?= 0.1,?= 0.5 3.2137 1.4902 0.8934 0.2528 0.0200 2.6707 1.3440 0.8699 0.2635 0.0230 ?= 0.1,?= 0.1,?= 1 0.3580 0.3299 0.2642 0.1875 0.0484 0.1302 0.0933 0.0670 0.0396 0.0104 ?= 0.1,?= 0.9,?= 1 0.3386 0.2362 0.1530 0.0743 0.0174 0.3246 0.2378 0.1694 0.1023 0.0597 ?= 0.5,?= 0.5,?= 1 0.7878 0.4347 0.2709 0.1015 0.0160 0.7621 0.4629 0.3235 0.1661 0.0654 ?= 0.9,?= 0.1,?= 1 14.3119 6.4062 3.7534 1.0360 0.0795 12.9766 6.1504 3.9367 1.1341 0.0898 ?= 0.1,?= 0.1,?= 5 7.4699 6.9473 6.3751 4.8796 0.9735 1.6871 1.2283 0.9607 0.5643 0.1710 ?= 0.1,?= 0.9,?= 5 6.4788 4.2740 2.8742 1.5911 0.3502 5.7193 4.0864 2.9201 2.0613 1.3153 ?= 0.5,?= 0.5,?= 5 15.0720 9.3556 6.3524 2.3258 0.3487 15.1315 9.4722 6.8444 3.3042 1.3018 ?= 0.9,?= 0.1,?= 5 325.6348 154.4483 89.1639 25.0749 2.1841 300.4573 163.8322 101.8301 30.2243 2.5295 ? ? ? M.Mohammadipour, 2009, Appendix 8.B 306 Appendix 8.B Impact of YW and CLS Estimates on Accuracy of Forecasts using MASE The comparison of YW and CLS estimates for INAR(1), INMA(1) and INARMA(1,1) in terms of their impact on forecast accuracy using MASE is presented in this appendix. Table 8.B-1 Forecast error comparison (YW and CLS) for INAR(1) series Parameters ??????/??????? ??????/??????? ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 0.1,?= 0.5 0.9888 0.9935 0.9975 0.9991 1.0000 0.9893 ?= 0.5,?= 0.5 0.9983 1.0009 1.0001 1.0008 1.0002 1.0116 ?= 0.9,?= 0.5 1.0837 1.0663 1.0483 1.0247 1.0038 1.1998 ?= 0.1,?= 1 0.9858 0.9955 0.9973 0.9989 0.9999 0.9810 ?= 0.5,?= 1 0.9942 0.9999 0.9993 1.0002 1.0001 1.0031 ?= 0.9,?= 1 1.0593 1.0461 1.0364 1.0149 1.0012 1.1401 ?= 0.1,?= 3 0.9916 0.9964 0.9979 0.9992 1.0000 0.9781 ?= 0.5,?= 3 0.9955 1.0007 1.0015 0.9998 1.0000 0.9931 ?= 0.9,?= 3 1.0462 1.0393 1.0265 1.0116 1.0005 1.1241 ?= 0.1,?= 5 0.9914 0.9955 0.9979 0.9992 1.0000 0.9825 ?= 0.5,?= 5 0.9956 0.9996 1.0001 1.0001 1.0000 1.0039 ?= 0.9,?= 5 1.0500 1.0411 1.0292 1.0120 1.0006 - Table 8.B-2 Forecast error comparison (YW and CLS) for INMA(1) series Parameters ??????/??????? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 1.0029 1.0012 1.0013 1.0004 1.0001 ?= 0.5,?= 0.5 1.0043 1.0053 1.0030 1.0016 1.0001 ?= 0.9,?= 0.5 1.0060 1.0074 1.0050 1.0027 0.9975 ?= 0.1,?= 1 1.0011 1.0019 1.0011 1.0009 1.0005 ?= 0.5,?= 1 1.0092 1.0075 1.0062 1.0053 1.0041 ?= 0.9,?= 1 1.0099 1.0077 1.0083 1.0025 0.9971 ?= 0.1,?= 3 1.0004 1.0010 1.0015 1.0008 1.0003 ?= 0.5,?= 3 1.0025 1.0049 1.0032 1.0048 1.0020 ?= 0.9,?= 3 1.0001 0.9991 1.0006 1.0035 1.0007 ?= 0.1,?= 5 0.9993 1.0005 1.0006 1.0005 1.0001 ?= 0.5,?= 5 1.0004 1.0001 1.0003 1.0020 1.0019 ?= 0.9,?= 5 0.9933 0.9960 0.9972 0.9980 1.0018 M.Mohammadipour, 2009, Appendix 8.B 307 Table 8.B-3 Comparison error comparison (YW and CLS) for INARMA(1,1) series Parameters ??????/??????? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.1,?= 0.5 0.9956 0.9984 1.0069 1.0072 1.0023 ?= 0.1,?= 0.9,?= 0.5 1.0093 1.0250 1.0251 1.0212 1.0072 ?= 0.5,?= 0.5,?= 0.5 1.0233 1.0419 1.0386 1.0246 1.0049 ?= 0.9,?= 0.1,?= 0.5 1.1859 1.1255 1.0907 1.0368 1.0065 ?= 0.1,?= 0.1,?= 1 1.0289 1.0316 1.0336 1.0293 1.0121 ?= 0.1,?= 0.9,?= 1 1.0245 1.0364 1.0361 1.0283 1.0095 ?= 0.5,?= 0.5,?= 1 1.0617 1.0558 1.0478 1.0281 1.0066 ?= 0.9,?= 0.1,?= 1 1.1955 1.1250 1.0889 1.0295 1.0030 ?= 0.1,?= 0.1,?= 5 1.0504 1.0474 1.0474 1.0473 1.0109 ?= 0.1,?= 0.9,?= 5 1.0577 1.0643 1.0662 1.0625 1.0351 ?= 0.5,?= 0.5,?= 5 1.0845 1.0751 1.0749 1.0415 1.0121 ?= 0.9,?= 0.1,?= 5 1.3024 1.1553 1.1013 1.0309 1.0022 M.Mohammadipour, 2009, Appendix 8.C 308 Appendix 8.C Croston-SBA Categorization for INAR(1), INMA(1) and INARMA(1,1) The following tables show that the Croston-SBA categorization generally holds for INAR(1), INMA(1), and INARMA(1,1) processes although it originally developed for i.i.d. processes. For INAR(1), when ?= 0.1,?= 0.5, ?= 0.5,?= 0.5, and ?= 0.1,?= 1, SBA should outperform Croston based on the corresponding p-value. Table 8.C-1 MSE of Croston and SBA with smoothing parameter 0.2 for INAR(1) series Parameters ?????????? ?????? ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.6782 0.6370 0.6204 0.6059 0.5888 0.6592 0.6248 0.6122 0.5986 0.5811 ?= 0.5,?= 0.5 1.1889 1.1333 1.1098 1.0575 1.0550 1.1558 1.1027 1.0866 1.0333 1.0302 ?= 0.9,?= 0.5 2.0661 2.1545 2.0588 2.0672 2.0163 2.3534 2.4621 2.3079 2.3119 2.2578 ?= 0.1,?= 1 1.2918 1.2387 1.2419 1.2364 1.2016 1.2686 1.2244 1.2305 1.2236 1.1879 ?= 0.5,?= 1 2.1306 2.0827 2.0617 2.0071 1.9784 2.1198 2.0755 2.0559 1.9977 1.9689 ?= 0.9,?= 1 4.3286 4.1355 4.1565 4.0094 3.9999 5.4842 5.2874 5.2252 5.0125 4.9994 ?= 0.1,?= 3 3.9531 3.8147 3.8414 3.7243 3.6340 4.0050 3.8476 3.8845 3.7627 3.6735 ?= 0.5,?= 3 6.0937 5.8541 5.7937 5.7145 5.6107 6.3761 6.1316 6.0812 5.9983 5.8763 ?= 0.9,?= 3 12.6744 12.5479 12.3444 12.0857 12.0343 21.6630 21.7462 21.4541 21.1452 21.0575 ?= 0.1,?= 5 6.7050 6.3945 6.3156 6.1690 6.0777 6.9397 6.6147 6.5153 6.3704 6.2710 ?= 0.5,?= 5 10.1928 9.6910 9.7351 9.3875 9.2467 11.1322 10.5665 10.6531 10.2523 10.0885 ?= 0.9,?= 5 20.4023 20.8507 20.9265 20.1313 19.9123 48.8346 47.8702 46.9309 46.1660 44.8889 Table 8.C-2 MSE of Croston and SBA with smoothing parameter 0.5 for INAR(1) series Parameters ?????????? ?????? ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.7352 0.7163 0.7050 0.6894 0.6711 0.6719 0.6559 0.6474 0.6334 0.6152 ?= 0.5,?= 0.5 1.1981 1.1722 1.1599 1.1137 1.1063 1.1030 1.0661 1.0646 1.0190 1.0096 ?= 0.9,?= 0.5 1.3146 1.3137 1.2846 1.2601 1.2360 3.1301 3.1514 2.9995 2.9740 2.9217 ?= 0.1,?= 1 1.4453 1.4006 1.3999 1.4021 1.3642 1.3495 1.3189 1.3243 1.3199 1.2818 ?= 0.5,?= 1 2.0781 2.0287 2.0136 1.9739 1.9389 2.1293 2.0815 2.0602 2.0147 1.9791 ?= 0.9,?= 1 2.7022 2.5724 2.5546 2.4636 2.4352 9.9577 9.5920 9.5507 9.0077 9.0156 ?= 0.1,?= 3 4.5453 4.4165 4.4184 4.2813 4.1936 4.8183 4.6422 4.6675 4.5205 4.4314 ?= 0.5,?= 3 5.8520 5.6100 5.5535 5.4866 5.3744 7.7889 7.4908 7.4554 7.3408 7.1728 ?= 0.9,?= 3 8.1248 7.7172 7.5610 7.3578 7.3587 68.5686 67.3196 66.2151 65.3240 64.8094 ?= 0.1,?= 5 7.7797 7.4433 7.3601 7.1823 7.0591 9.0522 8.7136 8.5879 8.3941 8.2367 ?= 0.5,?= 5 9.7481 9.2986 9.3367 9.0251 8.8548 15.7458 15.0617 15.1057 14.6590 14.3696 ?= 0.9,?= 5 12.8141 12.8380 12.6929 12.3732 12.1335 186.8114 180.7881 178.0403 175.0332 170.3159 M.Mohammadipour, 2009, Appendix 8.C 309 For INMA(1), when ?= 0.1,?= 0.5, ?= 0.5,?= 0.5, ?= 0.9,?= 0.5, and ?= 0.1,?= 1, SBA should outperform Croston based on the corresponding p-value. Table 8.C-3 MSE of Croston and SBA with smoothing parameter 0.2 for INMA(1) series Parameters ?????????? ?????? ?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.6682 0.6264 0.6169 0.5950 0.5826 0.6468 0.6148 0.6095 0.5876 0.5754 ?= 0.5,?= 0.5 0.9637 0.9103 0.8507 0.8495 0.8195 0.9285 0.8867 0.8324 0.8315 0.8030 ?= 0.9,?= 0.5 1.2077 1.1498 1.1031 1.0834 1.0584 1.1547 1.1164 1.0764 1.0562 1.0325 ?= 0.1,?= 1 1.3089 1.2502 1.2256 1.2002 1.1860 1.2915 1.2344 1.2120 1.1869 1.1729 ?= 0.5,?= 1 1.7781 1.6990 1.6832 1.6235 1.6365 1.7507 1.6803 1.6615 1.6031 1.6171 ?= 0.9,?= 1 2.2710 2.1981 2.1295 2.0714 2.0319 2.2540 2.1770 2.1101 2.0549 2.0131 ?= 0.1,?= 3 4.0167 3.8200 3.7213 3.6679 3.6364 4.0619 3.8592 3.7651 3.7018 3.6722 ?= 0.5,?= 3 5.1518 4.9393 4.9167 4.7959 4.6883 5.2691 5.0550 5.0298 4.9097 4.8008 ?= 0.9,?= 3 6.2893 6.1923 6.0164 5.8912 5.8008 6.5575 6.4307 6.2320 6.1184 6.0191 ?= 0.1,?= 5 6.5944 6.5473 6.2777 6.1716 6.0266 6.8133 6.7225 6.4675 6.3628 6.2156 ?= 0.5,?= 5 8.4587 8.1070 8.0842 7.9465 7.7757 8.9173 8.5772 8.5611 8.3625 0.5754 ?= 0.9,?= 5 10.6487 10.1183 10.1251 9.8462 9.6391 11.4393 10.8493 10.8578 10.5863 0.8030 Table 8.C-4 MSE of Croston and SBA with smoothing parameter 0.5 for INMA(1) series Parameters ?????????? ?????? ?=?? ?=?? ?= 48 ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=??? ?= 0.1,?= 0.5 0.7150 0.7091 0.6930 0.6787 0.6620 0.6528 0.6465 0.6407 0.6232 0.6088 ?= 0.5,?= 0.5 1.0583 1.0341 0.9870 0.9845 0.9506 0.9453 0.9228 0.8816 0.8793 0.8504 ?= 0.9,?= 0.5 1.3103 1.2889 1.2476 1.2304 1.2033 1.1635 1.1542 1.1219 1.0987 1.0740 ?= 0.1,?= 1 1.4580 1.4187 1.3924 1.3648 1.3498 1.3746 1.3317 1.3083 1.2818 1.2684 ?= 0.5,?= 1 1.9305 1.8791 1.8557 1.7775 1.7988 1.8414 1.7931 1.7662 1.6996 1.7163 ?= 0.9,?= 1 2.3641 2.3151 2.2397 2.1726 2.1380 2.3223 2.2708 2.2024 2.1439 2.1031 ?= 0.1,?= 3 4.6389 4.4053 4.2831 4.2405 4.2051 4.8967 4.6412 4.5360 4.4624 4.4243 ?= 0.5,?= 3 5.5047 5.3033 5.2630 5.1602 5.0283 6.3117 6.1039 6.0345 5.9233 5.7804 ?= 0.9,?= 3 6.4306 6.2870 6.1180 5.9501 5.8768 8.0736 7.8664 7.6395 7.4564 7.3493 ?= 0.1,?= 5 7.6307 7.6316 7.3059 7.1697 7.0108 8.8930 8.7858 8.4906 8.3299 8.1573 ?= 0.5,?= 5 9.0566 8.6844 8.6661 8.4742 8.3162 11.9868 11.6011 11.5405 11.2015 11.0221 ?= 0.9,?= 5 10.8019 10.1784 10.2150 9.9558 9.7360 15.8961 15.1546 15.1366 14.7591 14.4960 Finally, for INARMA(1,1), when ?= 0.1,?= 0.1,?= 0.5, ?= 0.1,?= 0.9,?= 0.5, ?= 0.5,?= 0.5,?= 0.5, and ?= 0.1,?= 0.1,?= 1, SBA should outperform Croston and the opposite is true for the rest. M.Mohammadipour, 2009, Appendix 8.C 310 Table 8.C-5 MSE of Croston and SBA with smoothing parameter 0.2 for INARMA(1,1) series Parameters ?????????? ?????? ?=?? ?=?? ?= 48 ?=?? ?=??? ?=?? ?=?? ?= 48 ?=?? ?=??? ?= 0.1,?= 0.1,?= 0.5 0.7572 0.7285 0.6975 0.6797 0.6672 0.7344 0.7133 0.6858 0.6691 0.6570 ?= 0.1,?= 0.9,?= 0.5 1.4658 1.3971 1.2976 1.2855 1.2758 1.4061 1.3529 1.2653 1.2548 1.2440 ?= 0.5,?= 0.5,?= 0.5 2.0666 1.9352 1.9751 1.8888 1.8780 2.0127 1.8915 1.9394 1.8472 1.8370 ?= 0.9,?= 0.1,?= 0.5 2.4911 2.4166 2.4541 2.3439 2.3694 2.9636 2.7476 2.7803 2.6402 2.6659 ?= 0.1,?= 0.1,?= 1 1.4875 1.4329 1.3832 1.3637 1.3459 1.4650 1.4128 1.3656 1.3466 1.3300 ?= 0.1,?= 0.9,?= 1 2.5633 2.6071 2.5802 2.4522 2.4170 2.5445 2.5750 2.5632 2.4316 2.3971 ?= 0.5,?= 0.5,?= 1 3.8122 3.5193 3.5507 3.4400 3.3913 3.8405 3.5307 3.5720 3.4446 3.4046 ?= 0.9,?= 0.1,?= 1 4.9377 4.9217 4.8844 4.9002 4.6510 6.4337 6.3152 6.2141 6.1640 5.8469 ?= 0.1,?= 0.1,?= 5 7.3304 7.0908 6.9961 6.8049 6.6737 7.6299 7.3524 7.2568 7.0638 6.9243 ?= 0.1,?= 0.9,?= 5 12.2581 11.8649 11.8106 11.4788 11.3795 13.3116 12.8960 12.7726 12.3983 12.2912 ?= 0.5,?= 0.5,?= 5 17.0179 16.6291 16.8486 16.3143 16.1399 19.2541 18.8518 18.9279 18.3672 18.1258 ?= 0.9,?= 0.1,?= 5 25.9712 23.8322 24.3298 24.2873 23.3176 57.2282 56.2972 55.9089 54.7003 53.6411 Table 8.C-6 MSE of Croston and SBA with smoothing parameter 0.5 for INARMA(1,1) series Parameters ?????????? ?????? ?=?? ?=?? ?= 48 ?=?? ?=??? ?=?? ?=?? ?= 48 ?=?? ?=??? ?= 0.1,?= 0.1,?= 0.5 0.8278 0.8255 0.7996 0.7832 0.7660 0.7468 0.7457 0.7237 0.7103 0.6956 ?= 0.1,?= 0.9,?= 0.5 1.5852 1.5845 1.4591 1.4463 1.4452 1.4111 1.3976 1.3085 1.2966 1.2900 ?= 0.5,?= 0.5,?= 0.5 1.9839 1.8595 1.9022 1.8264 1.8119 1.8514 1.7625 1.8132 1.7249 1.7129 ?= 0.9,?= 0.1,?= 0.5 1.4987 1.4757 1.4697 1.4072 1.4056 3.7822 3.5755 3.5684 3.4579 3.4478 ?= 0.1,?= 0.1,?= 1 1.6598 1.6111 1.5495 1.5359 1.5132 1.5622 1.5098 1.4599 1.4443 1.4251 ?= 0.1,?= 0.9,?= 1 2.6260 2.7332 2.6587 2.5459 2.5066 2.6141 2.6703 2.6471 2.5198 2.4867 ?= 0.5,?= 0.5,?= 1 3.3841 3.1385 3.1717 3.0929 3.0331 3.7330 3.4326 3.4858 3.3582 3.3188 ?= 0.9,?= 0.1,?= 1 3.0666 2.9571 2.9202 2.8722 2.7725 12.0092 11.4371 11.2427 10.9966 10.6574 ?= 0.1,?= 0.1,?= 5 8.2959 7.9922 7.8299 7.6759 7.5434 10.0345 9.6419 9.4752 9.2596 9.0982 ?= 0.1,?= 0.9,?= 5 12.2479 11.7657 11.7342 11.4066 11.3330 18.7567 18.1540 17.8894 17.4457 17.2565 ?= 0.5,?= 0.5,?= 5 14.8220 14.5978 14.7661 14.3685 14.1216 29.0148 28.5701 28.2932 27.7230 27.1435 ?= 0.9,?= 0.1,?= 5 15.6201 14.5525 14.5893 14.4396 13.9223 220.0655 216.2880 212.1152 207.3633 205.4631 M.Mohammadipour, 2009, Appendix 8.D 311 Appendix 8.D Comparing the Accuracy of INARMA Forecasts for all points in time and issue points In this appendix, the forecast accuracy of INARMA methods when all points in time are considered is compared to the case of issue points. Table 8.D-1 Forecast accuracy for all points in time and issue points for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? All points in time Issue Points All points in time Issue Points All points in time Issue Points All points in time Issue Points ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 0.0026 0.6614 1.1418 -0.1034 0.8816 0.8592 -0.0089 0.6286 1.0555 -0.0840 0.7817 0.8115 0.0052 0.6047 1.0237 -0.0491 0.7214 0.7881 0.0045 0.5822 0.9822 -0.0247 0.6703 0.7545 ?= 0.5,?= 0.5 -0.0026 0.9466 1.3290 -0.0241 1.2146 1.2289 -0.0089 0.8743 1.1807 -0.0316 1.0876 1.1100 0.0044 0.8532 1.1570 -0.0153 1.0373 1.0915 -0.0054 0.7834 1.0534 -0.0107 0.9473 0.9871 ?= 0.9,?= 0.5 0.0017 1.2082 1.3901 -0.0028 1.2116 1.3893 0.0042 1.1489 1.2894 0.0013 1.1517 1.2895 -0.0043 1.0811 1.2176 -0.0058 1.0841 1.2172 -0.0033 1.0164 1.1486 -0.0047 1.0200 1.1492 ?= 0.1,?= 1 -0.0149 1.2880 0.9376 -0.0962 1.4564 0.9781 -0.0059 1.2072 0.8496 -0.0582 1.2943 0.8580 0.0052 1.1967 0.8382 -0.0333 1.2759 0.8390 0.0029 1.1663 0.8057 -0.0132 1.2258 0.8090 ?= 0.5,?= 1 -0.0195 1.8846 1.0837 -0.0428 2.0093 1.1169 -0.0018 1.7244 1.0202 -0.0126 1.8430 1.0457 -0.0045 1.6745 1.0059 -0.0104 1.7796 1.0307 0.0026 1.5905 0.9618 0.0002 1.6749 0.9852 ?= 0.9,?= 1 -0.0121 2.5216 1.2401 -0.0125 2.5205 1.2400 0.0054 2.2655 1.1594 0.0053 2.2655 1.1594 -0.0146 2.1925 1.1262 -0.0147 2.1926 1.1262 -0.0105 2.0319 1.0686 -0.0107 2.0319 1.0686 ?= 0.1,?= 3 0.0100 3.9224 0.8544 -0.0101 3.9851 0.8781 -0.0176 3.6703 0.8228 -0.0242 3.7034 0.8448 -0.0010 3.6641 0.8136 -0.0101 3.7006 0.8353 0.0085 3.4701 0.7846 0.0050 3.4883 0.8019 ?= 0.5,?= 3 -0.0308 5.6926 1.0126 -0.0325 5.6986 1.0139 -0.0035 5.1557 0.9491 -0.0057 5.1609 0.9514 -0.0078 4.9742 0.9331 -0.0086 4.9801 0.9350 0.0143 4.7749 0.9160 0.0135 4.7793 0.9177 ?= 0.9,?= 3 -0.0906 7.5862 1.1509 -0.0906 7.5862 1.1509 -0.0298 6.7494 1.1026 -0.0298 6.7494 1.1026 -0.0243 6.4376 1.0658 -0.0243 6.4376 1.0658 -0.0093 6.0230 1.0205 -0.0093 6.0230 1.0205 ?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0335 6.6920 0.8580 0.0209 6.1869 0.8143 0.0179 6.1929 0.8176 0.0282 6.0095 0.7973 0.0260 6.0176 0.8016 0.0139 5.7529 0.7742 0.0133 5.7586 0.7778 ?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0121 9.3479 0.9843 0.0082 8.6581 0.9653 0.0082 8.6597 0.9655 0.0116 8.3583 0.9392 0.0116 8.3583 0.9393 -0.0034 7.8699 0.9011 -0.0035 7.8699 0.9012 ?= 0.9,?= 5 -0.0013 11.9986 1.1483 -0.0013 11.9986 1.1483 -0.0126 11.2051 1.0914 -0.0126 11.2051 1.0914 -0.0467 10.8985 1.0720 -0.0467 10.8985 1.0720 0.0185 10.1102 1.0272 0.0185 10.1102 1.0272 M.Mohammadipour, 2009, Appendix 8.D 312 Table 8.D-2 Forecast accuracy for all points in time and issue points for INMA(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? All points in time Issue Points All points in time Issue Points All points in time Issue Points All points in time Issue Points ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 -0.0031 0.6295 1.1297 -0.0392 0.7823 0.8062 0.0027 0.6035 1.0473 0.0117 0.7134 0.7866 0.0114 0.5993 1.0493 0.0249 0.6832 0.7875 0.0007 0.5702 0.9736 0.0480 0.6570 0.7351 ?= 0.5,?= 0.5 0.0395 0.8793 1.3407 0.1612 1.1706 1.1189 0.0279 0.8552 1.2527 0.1975 1.1200 1.0541 0.0229 0.7997 1.1494 0.1969 1.0123 0.9901 0.0031 0.7885 1.0948 0.2116 0.9847 0.9574 ?= 0.9,?= 0.5 0.0767 1.1019 1.5472 0.2912 1.4609 1.3594 0.0671 1.0609 1.3632 0.2997 1.3175 1.2363 0.0628 1.0229 1.2606 0.3007 1.2400 1.1211 0.0445 0.9878 1.1797 0.3103 1.1874 1.0683 ?= 0.1,?= 1 0.0234 1.2748 0.9163 0.0114 1.3964 0.9335 0.0000 1.2038 0.8547 0.0050 1.2923 0.8731 0.0082 1.1724 0.8365 0.0223 1.2557 0.8414 0.0048 1.1313 0.7957 0.0391 1.2009 0.7956 ?= 0.5,?= 1 0.0347 1.7554 1.0666 0.1211 1.9307 1.1132 0.0440 1.6455 1.0247 0.1414 1.7795 1.0523 0.0310 1.6074 1.0005 0.1429 1.7315 1.0136 0.0204 1.5302 0.9648 0.1512 1.6235 0.9718 ?= 0.9,?= 1 0.1249 2.2869 1.2485 0.2374 2.4432 1.2942 0.1025 2.1650 1.1687 0.2334 2.2741 1.1922 0.1148 2.0762 1.1223 0.2549 2.1785 1.1295 0.1116 1.9944 1.0920 0.2551 2.0664 1.0959 ?= 0.1,?= 3 0.0452 3.9039 0.8547 0.0376 3.9375 0.8748 0.0402 3.6622 0.8158 0.0408 3.6950 0.8347 0.0150 3.5353 0.7960 0.0174 3.5579 0.8157 0.0062 3.4237 0.7767 0.0129 3.4386 0.7954 ?= 0.5,?= 3 0.1260 5.2415 0.9971 0.1379 5.2750 1.0039 0.0781 4.9565 0.9450 0.0896 4.9647 0.9540 0.0929 4.8942 0.9424 0.1089 4.9067 0.9486 0.0608 4.6821 0.9036 0.0760 4.6843 0.9094 ?= 0.9,?= 3 0.2970 6.7422 1.1268 0.3041 6.7525 1.1288 0.3209 6.5993 1.1023 0.3271 6.6089 1.1045 0.2796 6.2599 1.0587 0.2873 6.2574 1.0596 0.2718 6.0678 1.0364 0.2801 6.0598 1.0369 ?= 0.1,?= 5 0.0838 6.4549 0.8431 0.0830 6.4710 0.8465 0.0132 6.2194 0.8182 0.0129 6.2335 0.8213 0.0226 5.9292 0.7937 0.0239 5.9364 0.7972 0.0111 5.7446 0.7806 0.0138 5.7500 0.7836 ?= 0.5,?= 5 0.1407 8.7278 0.9676 0.1413 8.7273 0.9679 0.1984 8.3020 0.9417 0.1991 8.3003 0.9416 0.1738 8.0993 0.9245 0.1751 8.1031 0.9250 0.1269 7.8404 0.9063 0.1281 7.8395 0.9067 ?= 0.9,?= 5 0.4387 11.6802 1.1864 0.4387 11.6802 1.1865 0.4081 10.9042 1.0713 0.4083 10.9074 1.0714 0.4804 10.7677 1.0895 0.4807 10.7679 1.0895 0.4479 10.1863 1.0242 0.4481 10.1860 1.0243 M.Mohammadipour, 2009, Appendix 8.D 313 Table 8.D-3 Forecast accuracy for all points in time and issue points for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? All points in time Issue Points All points in time Issue Points All points in time Issue Points All points in time Issue Points ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.1, ?= 0.5 -0.0007 0.7482 1.2163 -0.0700 1.0393 0.9354 -0.0066 0.7138 1.1186 -0.0382 0.9399 0.9074 0.0019 0.6777 1.0787 0.0039 0.8785 0.8779 -0.0047 0.6468 1.0076 0.0172 0.7936 0.8191 ?= 0.1,?= 0.9, ?= 0.5 0.0401 1.2112 1.4173 0.1570 1.5961 1.3111 0.0336 1.1340 1.2974 0.1877 1.4754 1.1898 0.0267 1.0734 1.1958 0.1686 1.3287 1.0998 0.0163 1.0429 1.1248 0.2050 1.2677 1.0215 ?= 0.5,?= 0.5, ?= 0.5 0.0485 1.5791 1.4864 0.1213 1.9469 1.3908 0.0209 1.4070 1.2666 0.0948 1.6876 1.2107 0.0445 1.3825 1.1924 0.1280 1.6513 1.1484 0.0261 1.2261 1.1137 0.1174 1.4511 1.0591 ?= 0.9,?= 0.1, ?= 0.5 0.0920 1.4003 1.4228 0.0903 1.4066 1.4232 0.0532 1.2843 1.3142 0.0514 1.2898 1.3152 0.0487 1.2226 1.2608 0.0482 1.2273 1.2634 0.0324 1.1110 1.1489 0.0317 1.1134 1.1492 ?= 0.1,?= 0.1, ?= 1 -0.0041 1.4908 0.9689 -0.0435 1.6925 1.0017 -0.0108 1.4146 0.9145 -0.0350 1.5629 0.9390 -0.0129 1.3358 0.8925 -0.0158 1.4591 0.9120 -0.0119 1.2833 0.8575 0.0018 1.3808 0.8725 ?= 0.1,?= 0.9, ?= 1 0.0192 2.3265 1.1462 0.0618 2.5072 1.1789 -0.0185 2.2569 1.0930 0.0507 2.4059 1.1171 0.0368 2.2224 1.0713 0.1105 2.3482 1.0744 0.0149 2.0900 1.0210 0.1107 2.1709 1.0172 ?= 0.5,?= 0.5, ?= 1 0.0296 3.1322 1.1914 0.0338 3.2529 1.2063 0.0250 2.7101 1.1033 0.0443 2.8128 1.1132 0.0409 2.6748 1.0798 0.0617 2.7683 1.0860 0.0220 2.4363 1.0245 0.0523 2.5181 1.0290 ?= 0.9,?= 0.1, ?= 1 0.0665 2.8324 1.2617 0.0665 2.8324 1.2617 0.0494 2.5623 1.1748 0.0492 2.5623 1.1747 0.0640 2.4301 1.1219 0.0640 2.4301 1.1219 0.0413 2.2588 1.0786 0.0413 2.2587 1.0786 ?= 0.1,?= 0.1, ?= 5 0.0279 7.4165 0.8824 0.0270 7.4271 0.8849 -0.0070 6.9501 0.8502 -0.0086 6.9563 0.8519 0.0011 6.7534 0.8307 -0.0008 6.7600 0.8323 0.0006 6.4253 0.8079 0.0011 6.4307 0.8101 ?= 0.1,?= 0.9, ?= 5 0.0436 11.2147 1.0506 0.0436 11.2147 1.0506 0.0580 10.5085 0.9941 0.0580 10.5096 0.9942 0.0521 10.1101 0.9589 0.0521 10.1101 0.9589 0.0606 9.7366 0.9338 0.0606 9.7366 0.9338 ?= 0.5,?= 0.5, ?= 5 0.1082 14.0754 1.0425 0.1082 14.0754 1.0425 0.1239 13.0227 1.0240 0.1239 13.0227 1.0240 0.1074 12.7060 0.9952 0.1074 12.7060 0.9952 0.1065 11.8259 0.9556 0.1065 11.8261 0.9556 ?= 0.9,?= 0.1, ?= 5 0.1214 13.8720 1.1795 0.1214 13.8720 1.1795 0.1895 12.4636 1.0932 0.1895 12.4636 1.0932 0.1569 12.0389 1.0813 0.1569 12.0389 1.0813 0.0906 11.2782 1.0311 0.0906 11.2782 1.0311 M.Mohammadipour, 2009, Appendix 8.E 314 Appendix 8.E Comparison of MASE of INARMA (known order) with Benchmarks In this appendix, the degree of improvement by INARMA over benchmarks, using the MASE measure, is presented. The results are for the case where all points in time are taken into account. Table 8.E-1 for INARMA(0,0) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.8905 0.9259 0.9350 0.9269 0.9610 0.9658 0.9499 0.9841 0.9882 0.9670 1.0007 1.0046 ?= 0.5 0.9622 0.9837 0.9868 0.9795 0.9987 1.0010 0.9852 1.0033 1.0053 0.9866 1.0053 1.0074 ?= 0.7 0.9836 0.9954 0.9961 0.9873 0.9955 0.9961 0.9894 0.9969 0.9974 0.9881 0.9960 0.9966 ?= 1 0.9773 0.9944 0.9953 0.9698 0.9851 0.9858 0.9591 0.9750 0.9758 0.9448 0.9605 0.9613 ?= 3 0.9766 0.9853 0.9844 0.9669 0.9760 0.9753 0.9631 0.9722 0.9715 0.9538 0.9622 0.9614 ?= 5 0.9756 0.9753 0.9726 0.9649 0.9623 0.9593 0.9631 0.9620 0.9591 0.9544 0.9535 0.9508 ?= 20 0.9792 0.9146 0.8987 0.9652 0.9056 0.8906 0.9633 0.9011 0.8857 0.9564 0.8957 0.8803 Table 8.E-2 for INMA(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9606 0.9810 0.9847 0.9775 0.9938 0.9956 0.9826 0.9981 0.9998 0.9867 1.0024 1.0041 ?= 0.5,?= 0.5 0.9459 0.9640 0.9665 0.9601 0.9751 0.9762 0.9660 0.9795 0.9805 0.9627 0.9772 0.9784 ?= 0.9,?= 0.5 0.9500 0.9717 0.9733 0.9453 0.9657 0.9679 0.9433 0.9617 0.9630 0.9243 0.9438 0.9453 ?= 0.1,?= 1 0.9757 0.9941 0.9953 0.9694 0.9900 0.9914 0.9603 0.9796 0.9808 0.9453 0.9657 0.9671 ?= 0.5,?= 1 0.9826 1.0020 1.0032 0.9828 1.0021 1.0034 0.9812 1.0010 1.0024 0.9796 1.0009 1.0026 ?= 0.9,?= 1 0.9955 1.0093 1.0098 0.9830 0.9974 0.9980 0.9786 0.9930 0.9935 0.9719 0.9864 0.9870 ?= 0.1,?= 3 0.9870 0.9934 0.9923 0.9740 0.9812 0.9802 0.9714 0.9796 0.9785 0.9675 0.9753 0.9743 ?= 0.5,?= 3 1.0073 1.0087 1.0060 0.9990 1.0010 0.9985 0.9977 0.9971 0.9943 0.9868 0.9878 0.9853 ?= 0.9,?= 3 1.0303 1.0191 1.0142 1.0296 1.0210 1.0164 1.0148 1.0087 1.0046 1.0089 1.0017 0.9973 ?= 0.1,?= 5 0.9883 0.9809 0.9771 0.9758 0.9756 0.9724 0.9690 0.9665 0.9632 0.9666 0.9625 0.9590 ?= 0.5,?= 5 1.0126 0.9957 0.9895 1.0091 0.9920 0.9855 0.9996 0.9825 0.9762 0.9906 0.9760 0.9700 ?= 0.9,?= 5 1.0493 1.0174 1.0084 1.0361 1.0119 1.0033 1.0252 1.0016 0.9933 1.0123 0.9875 0.9792 BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / M.Mohammadipour, 2009, Appendix 8.E 315 Table 8.E-3 for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 ?= 0.1,?= 0.5 0.9787 0.9505 0.9965 1.0121 0.3777 0.4554 0.9825 0.9381 0.9987 1.0027 1.0003 1.0329 0.9778 0.9312 0.9933 0.9957 0.9950 1.0273 0.9805 0.9342 0.9957 0.9979 0.9974 1.0279 ?= 0.5,?= 0.5 0.8685 0.8610 0.8837 0.9055 0.8851 0.8811 0.8619 0.8470 0.8772 0.8956 0.8784 0.8728 0.8595 0.8365 0.8745 0.8857 0.8755 0.8672 0.8435 0.8218 0.8598 0.8714 0.8610 0.8552 ?= 0.9,?= 0.5 0.7325 0.9403 0.6868 0.5770 0.6771 0.4419 0.7060 0.9252 0.6606 0.5598 0.6510 0.4228 0.6921 0.8938 0.6521 0.5511 0.6429 0.4191 0.6694 0.8741 0.6350 0.5394 0.6265 0.4093 ?= 0.1,?= 1 0.9966 0.9365 1.0161 0.9858 1.0175 1.0012 0.9812 0.9161 1.0012 0.9632 1.0025 0.9828 0.9710 0.9071 0.9895 0.9545 0.9907 0.9708 0.9570 0.8934 0.9773 0.9415 0.9787 0.9612 ?= 0.5,?= 1 0.9349 0.9482 0.9451 0.9571 0.9450 0.8936 0.9028 0.9227 0.9123 0.9256 0.9121 0.8622 0.8942 0.9079 0.9059 0.9180 0.9060 0.8574 0.8805 0.8954 0.8928 0.9039 0.8929 0.8467 ?= 0.9,?= 1 0.7581 0.9606 0.6670 0.4577 0.6496 0.3447 0.7313 0.9358 0.6460 0.4430 0.6295 0.3328 0.7165 0.9215 0.6402 0.4357 0.6239 0.3283 0.7004 0.9012 0.6266 0.4306 0.6111 0.3247 ?= 0.1,?= 3 0.9924 0.9297 0.9980 0.9202 0.9966 0.8996 0.9817 0.9136 0.9895 0.9117 0.9886 0.8982 0.9775 0.9152 0.9845 0.9078 0.9834 0.8899 0.9657 0.9028 0.9738 0.8988 0.9727 0.8794 ?= 0.5,?= 3 0.9629 0.9835 0.9480 0.8547 0.9427 0.7377 0.9354 0.9586 0.9263 0.8362 0.9214 0.7231 0.9233 0.9463 0.9104 0.8199 0.9053 0.7095 0.9121 0.9333 0.9007 0.8129 0.8958 0.7042 ?= 0.9,?= 3 0.7737 0.9664 0.5807 0.2813 0.5483 0.2114 0.7292 0.9333 0.5477 0.2692 0.5179 0.2021 0.7170 0.9195 0.5328 0.2631 0.5037 0.1975 0.7035 0.9023 0.5219 0.2572 0.4933 0.1930 ?= 0.1,?= 5 1.0011 0.9328 0.9919 0.8695 0.9879 0.8169 0.9861 0.9142 0.9803 0.8533 0.9764 0.8033 0.9765 0.9051 0.9729 0.8484 0.9692 0.7991 0.9651 0.8974 0.9627 0.8415 0.9594 0.7954 ?= 0.5,?= 5 0.9530 0.9735 0.9213 0.7596 0.9118 0.6289 0.9439 0.9638 0.9153 0.7520 0.9062 0.6239 0.9249 0.9464 0.8917 0.7361 0.8826 0.6100 0.9159 0.9355 0.8855 0.7255 0.8765 0.6026 ?= 0.9,?= 5 0.7668 0.9666 0.4736 0.2105 0.4388 0.1575 0.7357 0.9335 0.4633 0.2066 0.4301 0.1548 0.7213 0.9263 0.4623 0.2057 0.4294 0.1541 0.7078 0.9048 0.4473 0.1990 0.4151 0.1491 BenchmarkINARMA MASEMASE / M.Mohammadipour, 2009, Appendix 8.E 316 Table 8.E-4 for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 ?= 0.1,?= 0.1, ?= 0.5 0.9814 0.9497 1.0001 1.0118 1.0013 1.0305 0.9758 0.9275 0.9914 0.9938 0.9930 1.0190 0.9760 0.9256 0.9898 0.9893 0.9912 1.0136 0.9734 0.9184 0.9879 0.9850 0.9894 1.0135 ?= 0.1,?= 0.9, ?= 0.5 0.8861 0.8480 0.9101 0.9091 0.9105 0.9243 0.8839 0.8263 0.9048 0.8898 0.9065 0.9123 0.8869 0.8315 0.9066 0.8927 0.9082 0.9102 0.8753 0.8185 0.8963 0.8816 0.8980 0.9013 ?= 0.5,?= 0.5, ?= 0.5 0.8580 0.8754 0.8728 0.9157 0.8726 0.8639 0.8286 0.8488 0.8451 0.8887 0.8463 0.8393 0.8177 0.8307 0.8341 0.8720 0.8353 0.8269 0.7894 0.8027 0.8086 0.8470 0.8101 0.8075 ?= 0.9,?= 0.1, ?= 0.5 0.7329 0.9608 0.6755 0.5760 0.6648 0.4308 0.7079 0.9226 0.6649 0.5641 0.6555 0.4252 0.6793 0.8920 0.6412 0.5470 0.6324 0.4125 0.6602 0.8677 0.6275 0.5265 0.6189 0.3984 ?= 0.1,?= 0.1, ?= 1 0.9943 0.9404 1.0155 0.9913 1.0172 1.0070 0.9830 0.9236 1.0034 0.9746 1.0049 0.9951 0.9755 0.9196 0.9971 0.9694 0.9987 0.9868 0.9638 0.9072 0.9856 0.9579 0.9872 0.9777 ?= 0.1,?= 0.9, ?= 1 0.9421 0.9341 0.9531 0.9523 0.9530 0.9069 0.9172 0.9057 0.9325 0.9316 0.9332 0.9008 0.9145 0.9051 0.9283 0.9274 0.9287 0.8919 0.9127 0.9001 0.9282 0.9256 0.9288 0.8931 ?= 0.5,?= 0.5, ?= 1 0.8946 0.9514 0.8971 0.9190 0.8959 0.8149 0.8692 0.9220 0.8757 0.8987 0.8749 0.7999 0.8575 0.9107 0.8651 0.8892 0.8645 0.7934 0.8356 0.8852 0.8460 0.8682 0.8456 0.7767 ?= 0.9,?= 0.1, ?= 1 0.7502 0.9519 0.6610 0.4423 0.6431 0.3323 0.7118 0.9250 0.6280 0.4284 0.6115 0.3217 0.6924 0.9060 0.6162 0.4199 0.6005 0.3153 0.6693 0.8805 0.6004 0.4114 0.5854 0.3093 ?= 0.1,?= 0.1, ?= 5 1.0032 0.9442 0.9930 0.8617 0.9884 0.7962 0.9894 0.9338 0.9838 0.8610 0.9798 0.7976 0.9832 0.9325 0.9775 0.8558 0.9734 0.7890 0.9693 0.9142 0.9626 0.8407 0.9585 0.7795 ?= 0.1,?= 0.9, ?= 5 0.9532 0.9557 0.9246 0.7662 0.9161 0.6458 0.9347 0.9393 0.9065 0.7504 0.8981 0.6349 0.9226 0.9268 0.8982 0.7448 0.8902 0.6319 0.9176 0.9228 0.8923 0.7385 0.8843 0.6263 ?= 0.5,?= 0.5, ?= 5 0.9041 0.9718 0.8526 0.6689 0.8404 0.5331 0.8796 0.9427 0.8318 0.6528 0.8201 0.5215 0.8640 0.9250 0.8255 0.6533 0.8149 0.5235 0.8477 0.9035 0.8073 0.6340 0.7963 0.5079 ?= 0.9,?= 0.1, ?= 5 0.7328 0.9408 0.4704 0.2077 0.4368 0.1559 0.7219 0.9226 0.4469 0.1978 0.4142 0.1482 0.7002 0.9074 0.4446 0.1971 0.4125 0.1478 0.6810 0.6836 0.2741 0.6155 0.2058 0.1446 BenchmarkINARMA MASEMASE / M.Mohammadipour, 2009, Appendix 8.F 317 Appendix 8.F Comparison of six-step ahead MSE of INARMA (known order) with Benchmarks In this appendix the MSE of six-step ahead INARMA forecasts (using YW to estimate the parameters) is compared to the MSE of benchmark methods. Table 8.F-1 Six-step ahead for INARMA(0,0) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.8313 0.8826 0.8879 0.9378 0.9650 0.9675 0.9727 0.9891 0.9906 0.9789 0.9866 0.9873 ?= 0.5 0.9364 0.9695 0.9723 0.9747 0.9907 0.9917 0.9782 0.9891 0.9896 0.9648 0.9744 0.9747 ?= 0.7 0.9637 0.9882 0.9901 0.9751 0.9877 0.9880 0.9685 0.9787 0.9790 0.9581 0.9673 0.9674 ?= 1 0.9735 0.9894 0.9898 0.9675 0.9782 0.9780 0.9659 0.9740 0.9736 0.9468 0.9552 0.9549 ?= 3 0.9649 0.9548 0.9505 0.9478 0.9421 0.9382 0.9347 0.9295 0.9257 0.9206 0.9139 0.9100 ?= 5 0.9621 0.9455 0.9384 0.9419 0.9200 0.9125 0.9373 0.9146 0.9070 0.9143 0.8932 0.8860 ?= 20 0.9573 0.8218 0.7942 0.9389 0.8066 0.7794 0.9239 0.7887 0.7619 0.9142 0.7881 0.7619 Table 8.F-2 Six-step ahead for INMA(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9173 0.9540 0.9572 0.9675 0.9856 0.9866 0.9669 0.9813 0.9821 0.9591 0.9705 0.9710 ?= 0.5,?= 0.5 0.9211 0.9695 0.9737 0.9503 0.9794 0.9815 0.9420 0.9647 0.9660 0.9239 0.9456 0.9470 ?= 0.9,?= 0.5 0.9292 0.9808 0.9853 0.9536 0.9849 0.9870 0.9432 0.9667 0.9681 0.9108 0.9346 0.9361 ?= 0.1,?= 1 0.9682 0.9902 0.9912 0.9622 0.9762 0.9764 0.9499 0.9626 0.9626 0.9368 0.9496 0.9496 ?= 0.5,?= 1 0.9668 0.9801 0.9797 0.9496 0.9675 0.9677 0.9342 0.9488 0.9487 0.9093 0.9231 0.9229 ?= 0.9,?= 1 0.9688 0.9939 0.9944 0.9256 0.9393 0.9388 0.9006 0.9190 0.9189 0.8832 0.8986 0.8982 ?= 0.1,?= 3 0.9577 0.9557 0.9519 0.9377 0.9248 0.9200 0.9216 0.9125 0.9081 0.9124 0.9060 0.9019 ?= 0.5,?= 3 0.9567 0.9403 0.9342 0.9184 0.9130 0.9079 0.9028 0.8885 0.8827 0.8804 0.8729 0.8679 ?= 0.9,?= 3 0.9476 0.9341 0.9269 0.9073 0.8973 0.8909 0.8862 0.8747 0.8683 0.8664 0.8512 0.8445 ?= 0.1,?= 5 0.9542 0.9369 0.9292 0.9301 0.9093 0.9016 0.9177 0.8968 0.8893 0.8990 0.8739 0.8661 ?= 0.5,?= 5 0.9579 0.9239 0.9130 0.9150 0.8823 0.8721 0.8984 0.8680 0.8580 0.8774 0.8470 0.8372 ?= 0.9,?= 5 0.9455 0.8851 0.8710 0.9027 0.8651 0.8529 0.8832 0.8417 0.8293 0.8586 0.8193 0.8075 BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Appendix 8.F 318 Table 8.F-3 Six-step ahead with smoothing parameter 0.2 for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9290 0.9705 0.9745 0.9732 0.9921 0.9933 0.9714 0.9835 0.9841 0.9528 0.9655 0.9661 ?= 0.5,?= 0.5 1.0424 1.0900 1.0939 0.9670 0.9962 0.9981 0.9315 0.9587 0.9605 0.8780 0.9073 0.9094 ?= 0.9,?= 0.5 1.1764 1.1710 1.1607 1.0385 1.0363 1.0283 0.9826 0.9879 0.9810 0.8998 0.9066 0.9005 ?= 0.1,?= 1 0.9826 1.0046 1.0055 0.9705 0.9836 0.9836 0.9520 0.9622 0.9619 0.9356 0.9485 0.9486 ?= 0.5,?= 1 1.0263 1.0491 1.0492 0.9406 0.9667 0.9673 0.9170 0.9371 0.9371 0.8620 0.8837 0.8841 ?= 0.9,?= 1 1.1586 1.0883 1.0665 1.0187 0.9664 0.9480 0.9825 0.9356 0.9185 0.8863 0.8386 0.8223 ?= 0.1,?= 3 0.9748 0.9704 0.9664 0.9371 0.9315 0.9274 0.9255 0.9186 0.9143 0.9078 0.9008 0.8967 ?= 0.5,?= 3 0.9864 0.9747 0.9676 0.9131 0.9114 0.9055 0.8737 0.8627 0.8564 0.8393 0.8310 0.8251 ?= 0.9,?= 3 1.1322 0.8947 0.8454 1.0627 0.8301 0.7841 0.9961 0.7865 0.7450 0.8885 0.6957 0.6589 ?= 0.1,?= 5 0.9460 0.9188 0.9108 0.9225 0.8997 0.8919 0.9174 0.8917 0.8837 0.9026 0.8786 0.8708 ?= 0.5,?= 5 1.0105 0.9777 0.9643 0.9255 0.8918 0.8795 0.8862 0.8513 0.8394 0.8337 0.7990 0.7877 ?= 0.9,?= 5 1.1459 0.7273 0.6712 1.0230 0.6764 0.6242 0.9755 0.6680 0.6178 0.8781 0.6001 0.5553 Table 8.F-4 Six-step ahead with smoothing parameter 0.5 for INAR(1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.8777 0.9762 0.9872 0.8689 0.9483 0.9536 0.8531 0.9259 0.9297 0.8261 0.9035 0.9095 ?= 0.5,?= 0.5 0.9080 1.0391 1.0506 0.8214 0.9280 0.9327 0.7790 0.8829 0.8889 0.7437 0.8469 0.8534 ?= 0.9,?= 0.5 1.1361 0.9462 0.7806 1.0338 0.8704 0.7239 0.9877 0.8391 0.6981 0.9112 0.7704 0.6399 ?= 0.1,?= 1 0.8618 0.9320 0.9216 0.8344 0.8948 0.8832 0.8227 0.8788 0.8664 0.8092 0.8698 0.8589 ?= 0.5,?= 1 0.8658 0.9195 0.8916 0.7825 0.8503 0.8275 0.7727 0.8259 0.8007 0.7184 0.7748 0.7540 ?= 0.9,?= 1 1.1299 0.7420 0.5580 1.0106 0.6642 0.4991 0.9828 0.6461 0.4879 0.8961 0.5772 0.4317 ?= 0.1,?= 3 0.8300 0.8018 0.7387 0.7874 0.7614 0.7013 0.7713 0.7441 0.6846 0.7603 0.7329 0.6749 ?= 0.5,?= 3 0.8072 0.7399 0.6503 0.7424 0.6872 0.6053 0.7155 0.6538 0.5757 0.6844 0.6282 0.5531 ?= 0.9,?= 3 1.1316 0.3808 0.2443 1.0441 0.3545 0.2283 1.0112 0.3477 0.2253 0.9042 0.3097 0.2011 ?= 0.1,?= 5 0.7875 0.6921 0.6079 0.7670 0.6760 0.5899 0.7581 0.6655 0.5807 0.7516 0.6612 0.5766 ?= 0.5,?= 5 0.8134 0.6490 0.5302 0.7560 0.6022 0.4922 0.7214 0.5764 0.4725 0.6793 0.5413 0.4441 ?= 0.9,?= 5 1.1350 0.2570 0.1604 1.0247 0.2351 0.1454 0.9831 0.2347 0.1447 0.8907 0.2120 0.1311 BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Appendix 8.F 319 Table 8.F-5 Six-step ahead with smoothing parameter 0.2 for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9512 0.9912 0.9943 0.9824 1.0024 1.0038 0.9602 0.9795 0.9808 0.9486 0.9642 0.9650 ?= 0.1,?= 0.9,?= 0.5 0.9665 1.0085 1.0095 0.9345 0.9697 0.9722 0.9384 0.9654 0.9671 0.9016 0.9281 0.9299 ?= 0.5,?= 0.5,?= 0.5 0.9867 1.0228 1.0250 0.9133 0.9481 0.9506 0.8961 0.9309 0.9334 0.8562 0.8870 0.8890 ?= 0.9,?= 0.1,?= 0.5 1.0800 1.0416 1.0297 1.2237 1.2449 1.2379 1.1860 1.1924 1.1841 1.1816 1.1881 1.1794 ?= 0.1,?= 0.1,?= 1 1.0128 1.0355 1.0364 0.9989 1.0145 1.0147 0.9648 0.9791 0.9791 0.9254 0.9384 0.9384 ?= 0.1,?= 0.9,?= 1 0.9784 0.9922 0.9915 0.9274 0.9484 0.9485 0.9126 0.9302 0.9300 0.8686 0.8839 0.8835 ?= 0.5,?= 0.5,?= 1 0.9791 0.9998 0.9994 0.9318 0.9640 0.9648 0.8847 0.9072 0.9073 0.8367 0.8587 0.8588 ?= 0.9,?= 0.1,?= 1 1.2035 1.1125 1.0885 1.2078 1.1521 1.1302 1.2134 1.1574 1.1359 1.2180 1.1413 1.1183 ?= 0.1,?= 0.1,?= 5 0.9730 0.9560 0.9479 0.9283 0.9102 0.9024 0.9171 0.8995 0.8919 0.8933 0.8683 0.8601 ?= 0.1,?= 0.9,?= 5 0.9675 0.9382 0.9261 0.8961 0.8517 0.8392 0.8742 0.8333 0.8212 0.8486 0.8094 0.7974 ?= 0.5,?= 0.5,?= 5 0.9692 0.9024 0.8857 0.8892 0.8508 0.8372 0.8585 0.8164 0.8030 0.8123 0.7740 0.7614 ?= 0.9,?= 0.1,?= 5 1.3878 0.8918 0.8210 1.2591 0.8477 0.7841 1.3093 0.8560 0.7887 1.2348 0.8392 0.7758 Table 8.F-6 Six-step ahead with smoothing parameter 0.5 for INARMA(1,1) series (known order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.8927 0.9881 1.0203 0.8549 0.9433 0.9929 0.8271 0.9201 0.9682 0.8163 0.9023 0.9468 ?= 0.1,?= 0.9,?= 0.5 0.8253 0.9380 1.0095 0.7848 0.9007 0.9620 0.7870 0.8911 0.9425 0.7546 0.8594 0.9099 ?= 0.5,?= 0.5,?= 0.5 0.8752 0.9799 1.0064 0.7694 0.8795 0.9363 0.7577 0.8631 0.9183 0.7199 0.8149 0.8674 ?= 0.9,?= 0.1,?= 0.5 1.0323 0.8420 0.6669 1.2382 1.0821 0.8688 1.1947 1.0184 0.8128 1.1831 0.9947 0.7984 ?= 0.1,?= 0.1,?= 1 0.8833 0.9587 0.9878 0.8632 0.9266 0.9517 0.8292 0.8911 0.9180 0.7970 0.8552 0.8796 ?= 0.1,?= 0.9,?= 1 0.8156 0.8712 0.8932 0.7655 0.8308 0.8640 0.7558 0.8083 0.8436 0.7166 0.7668 0.7999 ?= 0.5,?= 0.5,?= 1 0.8055 0.8536 0.8870 0.7681 0.8332 0.8714 0.7397 0.7893 0.8140 0.6923 0.7404 0.7682 ?= 0.9,?= 0.1,?= 1 1.1949 0.7480 0.5465 1.2245 0.7934 0.5744 1.2326 0.8056 0.5853 1.2341 0.7862 0.5707 ?= 0.1,?= 0.1,?= 5 0.8034 0.7148 0.6574 0.7576 0.6714 0.6247 0.7533 0.6665 0.6185 0.7360 0.6413 0.5896 ?= 0.1,?= 0.9,?= 5 0.7717 0.6323 0.5574 0.7140 0.5658 0.4981 0.6934 0.5564 0.4883 0.6811 0.5392 0.4676 ?= 0.5,?= 0.5,?= 5 0.7894 0.5789 0.4843 0.7187 0.5541 0.4678 0.6910 0.5299 0.4482 0.6545 0.5015 0.4242 ?= 0.9,?= 0.1,?= 5 1.3669 0.3068 0.1871 1.2699 0.3015 0.1846 1.3210 0.2945 0.1798 1.2527 0.2946 0.1802 BenchmarkINARMA MSEMSE / BenchmarkINARMA MSEMSE / M.Mohammadipour, 2009, Appendix 8.G 320 Appendix 8.G Comparison of all-INAR(1) and all-INARMA(1,1) In this appendix, the forecast accuracy of all-INAR(1) and all-INARMA(1,1) is compared. The results are presented for the case of INAR(1), INMA(1) and INARMA(1,1) series. The corresponding results for an INARMA(0,0) series can be found from Table ?8-32 and Table ?8-43. Table 8.G-1 Accuracy of forecasts by all-INAR(1) and all-INARMA(1,1) approaches for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 0.0026 0.6614 1.1418 -0.0008 0.6699 1.1275 -0.0089 0.6286 1.0555 0.0049 0.6447 1.0710 0.0052 0.6047 1.0237 0.0111 0.6389 1.0340 0.0045 0.5822 0.9822 0.0010 0.5857 0.9845 ?= 0.5,?= 0.5 -0.0026 0.9466 1.3290 0.0130 1.0056 1.3278 -0.0089 0.8743 1.1807 0.0039 0.9436 1.2602 0.0044 0.8532 1.1570 0.0041 0.9131 1.1864 -0.0054 0.7834 1.0534 -0.0033 0.8186 1.0727 ?= 0.9,?= 0.5 0.0017 1.2082 1.3901 0.0347 1.2681 1.4405 0.0042 1.1489 1.2894 0.0344 1.1681 1.3181 -0.0043 1.0811 1.2176 0.0066 1.0919 1.2150 -0.0033 1.0164 1.1486 0.0206 1.0235 1.1707 ?= 0.1,?= 1 -0.0149 1.2880 0.9376 -0.0152 1.3755 0.9455 -0.0059 1.2072 0.8496 -0.0028 1.2778 0.8819 0.0052 1.1967 0.8382 -0.0066 1.1947 0.8427 0.0029 1.1663 0.8057 0.0045 1.1875 0.8235 ?= 0.5,?= 1 -0.0195 1.8846 1.0837 -0.0092 1.9908 1.1188 -0.0018 1.7244 1.0202 0.0050 1.8491 1.0640 -0.0045 1.6745 1.0059 -0.0046 1.7735 1.0251 0.0026 1.5905 0.9618 0.0052 1.6469 0.9944 ?= 0.9,?= 1 -0.0121 2.5216 1.2401 0.0544 2.4325 1.2304 0.0054 2.2655 1.1594 0.0378 2.3301 1.1748 -0.0146 2.1925 1.1262 0.0484 2.1853 1.1423 -0.0105 2.0319 1.0686 0.0312 2.0343 1.0747 ?= 0.1,?= 3 0.0100 3.9224 0.8544 -0.0275 4.1292 0.8818 -0.0176 3.6703 0.8228 -0.0177 3.7457 0.8236 -0.0010 3.6641 0.8136 -0.0144 3.6442 0.8115 0.0085 3.4701 0.7846 -0.0222 3.4929 0.7828 ?= 0.5,?= 3 -0.0308 5.6926 1.0126 0.0237 5.8020 1.0279 -0.0035 5.1557 0.9491 -0.0291 5.4831 0.9848 -0.0078 4.9742 0.9331 -0.0027 5.2166 0.9710 0.0143 4.7749 0.9160 0.0268 4.9748 0.9319 ?= 0.9,?= 3 -0.0906 7.5862 1.1509 0.1097 7.3168 1.1422 -0.0298 6.7494 1.1026 0.1077 6.9143 1.1046 -0.0243 6.4376 1.0658 0.0627 6.5204 1.0742 -0.0093 6.0230 1.0205 0.0612 6.1440 1.0290 ?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0124 6.5975 0.8522 0.0209 6.1869 0.8143 0.0016 6.3370 0.8206 0.0282 6.0095 0.7973 -0.0078 6.2455 0.8197 0.0139 5.7529 0.7742 -0.0063 5.8121 0.7805 ?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0798 9.5203 1.0114 0.0082 8.6581 0.9653 0.0050 8.8544 0.9603 0.0116 8.3583 0.9392 0.0325 8.4735 0.9460 -0.0034 7.8699 0.9011 -0.0003 7.9638 0.9095 ?= 0.9,?= 5 -0.0013 11.9986 1.1483 0.0773 12.2386 1.1385 -0.0126 11.2051 1.0914 0.1196 11.3739 1.0820 -0.0467 10.8985 1.0720 0.1021 10.9289 1.0679 0.0185 10.1102 1.0272 0.0956 10.1452 1.0232 M.Mohammadipour, 2009, Appendix 8.G 321 Table 8.G-2 Accuracy of forecasts by all-INAR(1) and all-INARMA(1,1) approaches for INMA(1) series Parameters ?=?? ?=?? ?=?? ?=?? All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.5 -0.0102 0.6916 1.1405 -0.0056 0.6500 1.1831 0.0016 0.6215 1.0690 -0.0038 0.6155 1.0589 -0.0084 0.5981 1.0149 -0.0071 0.6070 1.0025 0.00007 0.5687 0.9692 0.0063 0.5733 0.9809 ?= 0.5,?= 0.5 -0.0058 0.8285 1.2415 0.0205 0.8681 1.3428 0.0085 0.7828 1.1608 -0.0151 0.8047 1.1628 0.0033 0.7599 1.1155 0.0011 0.7815 1.1069 0.0038 0.6959 1.0381 -0.0038 0.7423 1.0637 ?= 0.9,?= 0.5 -0.0324 0.9442 1.2536 0.0430 1.0089 1.3941 -0.0052 0.8721 1.2436 0.0352 0.9383 1.2649 -0.0089 0.8149 1.1243 0.0122 0.9436 1.1917 -0.0063 0.7802 1.0677 0.0250 0.9208 1.1448 ?= 0.1,?= 1 -0.0005 1.3118 0.8950 -0.0202 1.3447 0.9356 -0.0078 1.2120 0.8617 -0.0154 1.2458 0.8487 0.0034 1.1882 0.8419 -0.0017 1.1956 0.8443 -0.0013 1.1519 0.8002 -0.0016 1.1525 0.8117 ?= 0.5,?= 1 -0.0271 1.6581 1.0424 0.0060 1.6743 1.0374 -0.0235 1.5574 0.9788 -0.0128 1.5905 0.9873 -0.0039 1.4772 0.9546 0.0030 1.5879 0.9894 -0.0047 1.4121 0.9168 -0.0077 1.4733 0.9413 ?= 0.9,?= 1 -0.0443 1.9014 1.1317 0.0145 1.9996 1.1350 -0.0203 1.7164 1.0562 0.0180 1.9324 1.0960 -0.0292 1.6602 1.0051 0.0165 1.8242 1.0545 -0.0127 1.5469 0.9562 0.0184 1.7825 1.0406 ?= 0.1,?= 3 0.0169 3.8869 0.8397 -0.0153 3.8953 0.8642 -0.0190 3.6392 0.8194 0.0022 3.7206 0.8211 -0.0208 3.4924 0.7922 0.0041 3.6216 0.7983 0.0136 3.4409 0.7838 -0.0215 3.5003 0.7873 ?= 0.5,?= 3 -0.0436 4.8555 0.9562 -0.0306 4.8945 0.9557 -0.0302 4.6371 0.9278 -0.0137 4.7351 0.9278 -0.0170 4.4562 0.8843 -0.0013 4.6193 0.9143 -0.0039 4.2607 0.8655 0.0061 4.3870 0.8783 ?= 0.9,?= 3 -0.1126 5.4157 1.0076 0.0763 5.8039 1.0545 -0.0603 5.1071 0.9687 0.0192 5.5915 1.0116 -0.0481 4.8830 0.9420 0.0118 5.2638 0.9726 -0.0213 4.7140 0.9137 0.0439 5.1879 0.9612 ?= 0.1,?= 5 -0.0330 6.3471 0.8369 0.0108 6.5867 0.8570 -0.0025 6.1832 0.8223 -0.0108 6.2327 0.8241 -0.0221 6.0512 0.7981 -0.0045 6.0357 0.7906 -0.0054 5.6957 0.7776 -0.0103 5.7236 0.7712 ?= 0.5,?= 5 -0.0071 8.0533 0.9474 -0.0335 8.3565 0.9718 -0.0317 7.5846 0.8934 0.0231 7.7461 0.9228 -0.0207 7.4066 0.8805 0.0136 7.6924 0.8954 -0.0166 6.8773 0.8510 0.0160 7.2509 0.8698 ?= 0.9,?= 5 -0.1464 9.3220 1.0338 0.0927 9.8826 1.0531 -0.0607 8.7192 0.9763 0.0643 8.8871 0.9798 -0.0626 8.1938 0.9370 0.0683 8.9059 0.9673 -0.0218 7.7859 0.9012 0.0524 8.4545 0.9443 M.Mohammadipour, 2009, Appendix 8.G 322 Table 8.G-3 Accuracy of forecasts by all-INAR(1) and all-INARMA(1,1) approaches for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) All-INAR(1) All-INARMA(1,1) ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ?= 0.1,?= 0.1, ?= 0.5 -0.0078 0.7157 1.1853 -0.0007 0.7482 1.2163 0.0050 0.6776 1.0914 -0.0066 0.7138 1.1186 -0.0037 0.6789 1.0606 0.0019 0.6777 1.0787 0.0034 0.6319 1.0015 -0.0047 0.6468 1.0076 ?= 0.1,?= 0.9, ?= 0.5 -0.0336 1.1234 1.3333 0.0401 1.2112 1.4173 -0.0166 1.0234 1.1892 0.0336 1.1340 1.2974 -0.0108 0.9883 1.1387 0.0267 1.0734 1.1958 -0.0082 0.9030 1.0507 0.0163 1.0429 1.1248 ?= 0.5,?= 0.5, ?= 0.5 -0.0397 1.4167 1.3403 0.0485 1.5791 1.4864 -0.0108 1.3213 1.2244 0.0209 1.4070 1.2666 -0.0025 1.2331 1.1576 0.0445 1.3825 1.1924 -0.0109 1.1697 1.0806 0.0261 1.2261 1.1137 ?= 0.9,?= 0.1, ?= 0.5 0.0022 1.3138 1.3528 0.0920 1.4003 1.4228 -0.0171 1.2480 1.2641 0.0532 1.2843 1.3142 -0.0192 1.2192 1.2223 0.0487 1.2226 1.2608 0.0009 1.1111 1.1593 0.0324 1.1110 1.1489 ?= 0.1,?= 0.1, ?= 1 -0.0092 1.4535 0.9748 -0.0041 1.4908 0.9689 0.0050 1.3592 0.8997 -0.0108 1.4146 0.9145 -0.0022 1.3239 0.8900 -0.0129 1.3358 0.8925 0.0043 1.2502 0.8470 -0.0119 1.2833 0.8575 ?= 0.1,?= 0.9, ?= 1 -0.0421 2.2562 1.1278 0.0192 2.3265 1.1462 -0.0379 1.9904 1.0257 -0.0185 2.2569 1.0930 -0.0301 1.9530 1.0063 0.0368 2.2224 1.0713 -0.0113 1.8374 0.9588 0.0149 2.0900 1.0210 ?= 0.5,?= 0.5, ?= 1 -0.0791 2.9429 1.1710 0.0296 3.1322 1.1914 -0.0416 2.6021 1.0770 0.0250 2.7101 1.1033 -0.0334 2.4496 1.0386 0.0409 2.6748 1.0798 -0.0145 2.3081 0.9957 0.0220 2.4363 1.0245 ?= 0.9,?= 0.1, ?= 1 -0.0718 2.7062 1.2374 0.0665 2.8324 1.2617 -0.0625 2.4894 1.1512 0.0494 2.5623 1.1748 -0.0466 2.3962 1.1273 0.0640 2.4301 1.1219 -0.0156 2.2173 1.0745 0.0413 2.2588 1.0786 ?= 0.1,?= 0.1, ?= 5 -0.0116 7.2720 0.8837 0.0279 7.4165 0.8824 -0.0119 6.7888 0.8526 -0.0070 6.9501 0.8502 0.0285 6.6017 0.8402 0.0011 6.7534 0.8307 -0.0169 6.2483 0.8011 0.0006 6.4253 0.8079 ?= 0.1,?= 0.9, ?= 5 -0.1451 11.3162 1.0522 0.0436 11.2147 1.0506 -0.1048 10.1377 0.9675 0.0580 10.5085 0.9941 -0.0779 9.7491 0.9463 0.0521 10.1101 0.9589 -0.0438 9.1660 0.9021 0.0606 9.7366 0.9338 ?= 0.5,?= 0.5, ?= 5 -0.1845 14.2594 1.0796 0.1082 14.0754 1.0425 -0.1399 12.7866 1.0133 0.1239 13.0227 1.0240 -0.0804 12.2174 0.9767 0.1074 12.7060 0.9952 -0.0521 11.5310 0.9499 0.1065 11.8259 0.9556 ?= 0.9,?= 0.1, ?= 5 -0.3143 14.2380 1.1727 0.1214 13.8720 1.1795 -0.1687 12.8863 1.1125 0.1895 12.4636 1.0932 -0.0959 12.1338 1.0723 0.1569 12.0389 1.0813 -0.0613 11.0139 1.0149 0.0906 11.2782 1.0311 M.Mohammadipour, 2009, Appendix 8.H 323 Appendix 8.H Comparison of MASE of INARMA (unknown order) with Benchmarks In this appendix, the degree of improvement by using all-INAR(1) over benchmarks, in terms of the MASE is presented. The results are for the case where all points in time are taken into account. The results for INAR(1) series are the same as the case where the order is known (Table 8.E-3). Table 8.H-1 for INARMA(0,0) series (unknown order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.8989 0.9346 0.9438 0.9497 0.9846 0.9895 0.9533 0.9877 0.9918 0.9647 0.9983 1.0023 ?= 0.5 0.9777 0.9995 1.0027 0.9877 1.0071 1.0094 1.0007 1.0192 1.0212 1.0002 1.0192 1.0213 ?= 0.7 0.9992 1.0112 1.0119 0.9941 1.0024 1.0030 1.0107 1.0185 1.0189 0.9813 0.9892 0.9898 ?= 1 1.0102 1.0278 1.0287 0.9808 0.9963 0.9970 0.9691 0.9851 0.9860 0.9644 0.9804 0.9813 ?= 3 0.9967 1.0056 1.0047 0.9643 0.9734 0.9727 0.9577 0.9668 0.9660 0.9472 0.9555 0.9548 ?= 5 0.9688 0.9684 0.9658 0.9527 0.9502 0.9472 0.9745 0.9734 0.9704 0.9655 0.9646 0.9618 ?= 20 1.0024 0.9362 0.9199 0.9682 0.9083 0.8932 0.9815 0.9182 0.9025 0.9589 0.8980 0.8826 Table 8.H-2 for INMA(1) series (unknown order) Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9698 0.9904 0.9941 0.9978 1.0144 1.0163 0.9504 0.9654 0.9670 0.9823 0.9978 0.9996 ?= 0.5,?= 0.5 0.8759 0.8927 0.8950 0.8897 0.9036 0.9045 0.9375 0.9506 0.9516 0.9129 0.9266 0.9277 ?= 0.9,?= 0.5 0.7697 0.7873 0.7886 0.8624 0.8810 0.8830 0.8413 0.8577 0.8589 0.8366 0.8542 0.8556 ?= 0.1,?= 1 0.9530 0.9710 0.9722 0.9773 0.9981 0.9995 0.9665 0.9859 0.9871 0.9507 0.9711 0.9725 ?= 0.5,?= 1 0.9603 0.9792 0.9804 0.9388 0.9572 0.9585 0.9362 0.9551 0.9564 0.9309 0.9511 0.9527 ?= 0.9,?= 1 0.9024 0.9149 0.9153 0.8884 0.9014 0.9019 0.8764 0.8893 0.8898 0.8510 0.8637 0.8642 ?= 0.1,?= 3 0.9696 0.9759 0.9749 0.9783 0.9856 0.9845 0.9668 0.9749 0.9738 0.9763 0.9842 0.9832 ?= 0.5,?= 3 0.9660 0.9673 0.9647 0.9809 0.9827 0.9803 0.9362 0.9357 0.9330 0.9452 0.9461 0.9437 ?= 0.9,?= 3 0.9213 0.9113 0.9069 0.9048 0.8973 0.8932 0.9029 0.8975 0.8938 0.8894 0.8831 0.8792 ?= 0.1,?= 5 0.9810 0.9737 0.9699 0.9807 0.9804 0.9773 0.9744 0.9719 0.9686 0.9629 0.9588 0.9553 ?= 0.5,?= 5 0.9914 0.9749 0.9688 0.9574 0.9411 0.9349 0.9520 0.9357 0.9298 0.9302 0.9164 0.9108 ?= 0.9,?= 5 0.9143 0.8865 0.8787 0.9442 0.9222 0.9143 0.8817 0.8614 0.8543 0.8907 0.8689 0.8616 BenchmarkINARMA MASEMASE / BenchmarkINARMA MASEMASE / M.Mohammadipour, 2009, Appendix 8.H 324 Table 8.H-3 for INARMA(1,1) series (unknown order) Parameters ?=?? ?=?? ?=?? ?=?? Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 Cros 0.2 Cros 0.5 SBA 0.2 SBA 0.5 SBJ 0.2 SBJ 0.5 ?= 0.1,?= 0.1,?= 0.5 0.9564 0.9255 0.9746 0.9860 0.9758 1.0042 0.9520 0.9050 0.9673 0.9696 0.9688 0.9943 0.9596 0.9101 0.9732 0.9727 0.9745 0.9966 0.9675 0.9129 0.9820 0.9791 0.9834 1.0073 ?= 0.1,?= 0.9,?= 0.5 0.8336 0.7977 0.8562 0.8552 0.8565 0.8695 0.8102 0.7574 0.8293 0.8156 0.8309 0.8362 0.8445 0.7918 0.8633 0.8501 0.8648 0.8667 0.8176 0.7645 0.8372 0.8235 0.8389 0.8419 ?= 0.5,?= 0.5,?= 0.5 0.7737 0.7894 0.7870 0.8257 0.7868 0.7790 0.8010 0.8205 0.8170 0.8591 0.8181 0.8113 0.7938 0.8065 0.8098 0.8465 0.8109 0.8028 0.7659 0.7789 0.7845 0.8219 0.7860 0.7835 ?= 0.9,?= 0.1,?= 0.5 0.6969 0.9135 0.6423 0.5477 0.6321 0.4096 0.6809 0.8874 0.6396 0.5426 0.6305 0.4090 0.6586 0.8647 0.6216 0.5303 0.6131 0.3999 0.6662 0.8756 0.6332 0.5312 0.6245 0.4020 ?= 0.1,?= 0.1,?= 1 1.0003 0.9461 1.0217 0.9973 1.0234 1.0131 0.9671 0.9087 0.9872 0.9589 0.9887 0.9790 0.9728 0.9171 0.9943 0.9667 0.9959 0.9841 0.9520 0.8961 0.9736 0.9462 0.9751 0.9657 ?= 0.1,?= 0.9,?= 1 0.9270 0.9192 0.9378 0.9370 0.9377 0.8924 0.8607 0.8499 0.8751 0.8742 0.8757 0.8454 0.8591 0.8502 0.8720 0.8711 0.8724 0.8377 0.8571 0.8453 0.8716 0.8692 0.8722 0.8387 ?= 0.5,?= 0.5,?= 1 0.8793 0.9351 0.8818 0.9033 0.8806 0.8009 0.8485 0.9001 0.8548 0.8773 0.8540 0.7808 0.8247 0.8759 0.8321 0.8553 0.8315 0.7631 0.8122 0.8603 0.8222 0.8438 0.8218 0.7548 ?= 0.9,?= 0.1,?= 1 0.7358 0.9336 0.6482 0.4337 0.6307 0.3259 0.6975 0.9064 0.6154 0.4198 0.5992 0.3153 0.6958 0.9104 0.6192 0.4219 0.6034 0.3168 0.6667 0.8771 0.5981 0.4099 0.5832 0.3081 ?= 0.1,?= 0.1,?= 5 1.0047 0.9456 0.9945 0.8630 0.9898 0.7973 0.9922 0.9364 0.9866 0.8635 0.9826 0.7998 0.9944 0.9432 0.9887 0.8656 0.9845 0.7980 0.9611 0.9065 0.9545 0.8336 0.9504 0.7730 ?= 0.1,?= 0.9,?= 5 0.9546 0.9572 0.9260 0.7674 0.9175 0.6468 0.9096 0.9142 0.8823 0.7304 0.8741 0.6179 0.9105 0.9147 0.8864 0.7350 0.8785 0.6236 0.8864 0.8915 0.8620 0.7134 0.8543 0.6051 ?= 0.5,?= 0.5,?= 5 0.9363 1.0064 0.8830 0.6927 0.8703 0.5520 0.8704 0.9329 0.8232 0.6460 0.8115 0.5161 0.8479 0.9078 0.8102 0.6411 0.7997 0.5138 0.8426 0.8981 0.8025 0.6302 0.7916 0.5048 ?= 0.9,?= 0.1,?= 5 0.7286 0.9354 0.4676 0.2065 0.4342 0.1550 0.7346 0.9389 0.4548 0.2012 0.4215 0.1508 0.6944 0.8999 0.4409 0.1955 0.4090 0.1465 0.6703 0.8707 0.4273 0.1897 0.3968 0.1423 BenchmarkINARMA MASEMASE / M.Mohammadipour, 2009, Appendix 8.I 325 Appendix 8.I Comparison of MASE of INARMA with Benchmarks for Lead Time Forecasts In this appendix, the lead time forecasts of an all-INAR(1) method are compared to those of the benchmarks methods in terms of MASE. The results are presented for INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1) series and include both cases of ?= 3 and ?= 6. Table 8.I-1 of lead-time forecasts for INARMA(0,0) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.8692 0.9131 0.9178 0.9420 0.9748 0.9779 0.9692 0.9899 0.9917 0.9603 0.9773 0.9785 ?= 0.5 0.9484 0.9840 0.9874 0.9739 0.9993 1.0013 0.9742 0.9990 1.0010 0.9559 0.9827 0.9849 ?= 0.7 0.9755 1.0034 1.0054 0.9722 0.9968 0.9983 0.9622 0.9846 0.9860 0.9364 0.9581 0.9593 ?= 1 0.9824 0.9999 1.0000 0.9629 0.9819 0.9824 0.9516 0.9714 0.9719 0.9247 0.9481 0.9491 ?= 3 0.9624 0.9560 0.9512 0.9303 0.9263 0.9218 0.9121 0.9100 0.9058 0.8984 0.8963 0.8920 ?= 5 0.9567 0.9282 0.9184 0.9299 0.9056 0.8967 0.9112 0.8863 0.8775 0.8923 0.8710 0.8625 ?= 20 0.9552 0.7915 0.7606 0.9198 0.7657 0.7352 0.9044 0.7575 0.7286 0.8822 0.7413 0.7129 Table 8.I-2 of lead-time forecasts for INMA(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9463 0.9858 0.9897 0.9682 0.9978 1.0003 0.9644 0.9914 0.9937 0.9506 0.9789 0.9812 ?= 0.5,?= 0.5 0.9748 1.0102 1.0134 0.9535 0.9838 0.9864 0.9443 0.9735 0.9759 0.9173 0.9462 0.9486 ?= 0.9,?= 0.5 0.9756 1.0186 1.0225 0.9587 0.9907 0.9933 0.9426 0.9727 0.9751 0.9066 0.9363 0.9388 ?= 0.1,?= 1 0.9854 1.0073 1.0082 0.9612 0.9833 0.9843 0.9455 0.9688 0.9699 0.9249 0.9493 0.9505 ?= 0.5,?= 1 0.9848 1.0053 1.0061 0.9577 0.9769 0.9774 0.9437 0.9646 0.9654 0.9120 0.9354 0.9364 ?= 0.9,?= 1 0.9974 1.0177 1.0183 0.9619 0.9824 0.9830 0.9447 0.9630 0.9632 0.9163 0.9376 0.9382 ?= 0.1,?= 3 0.9688 0.9579 0.9523 0.9391 0.9423 0.9386 0.9241 0.9209 0.9166 0.9010 0.9011 0.8972 ?= 0.5,?= 3 0.9849 0.9757 0.9701 0.9481 0.9412 0.9358 0.9369 0.9330 0.9281 0.9062 0.9027 0.8980 ?= 0.9,?= 3 0.9993 0.9771 0.9698 0.9592 0.9415 0.9344 0.9437 0.9346 0.9286 0.9103 0.9009 0.8949 ?= 0.1,?= 5 0.9686 0.9468 0.9378 0.9397 0.9124 0.9029 0.9151 0.8965 0.8882 0.9009 0.8794 0.8708 ?= 0.5,?= 5 0.9822 0.9528 0.9423 0.9451 0.9115 0.9014 0.9356 0.9060 0.8959 0.9077 0.8796 0.8701 ?= 0.9,?= 5 0.9935 0.9527 0.9404 0.9514 0.9110 0.8990 0.9358 0.9018 0.8906 0.9120 0.8763 0.8653 BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 3?l M.Mohammadipour, 2009, Appendix 8.I 326 Table 8.I-3 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9691 1.0016 1.0046 0.9605 0.9901 0.9927 0.9635 0.9896 0.9918 0.9461 0.9730 0.9753 ?= 0.5,?= 0.5 0.9392 0.9627 0.9644 0.9229 0.9469 0.9489 0.9013 0.9267 0.9288 0.8690 0.8978 0.9002 ?= 0.9,?= 0.5 0.8990 0.8433 0.8308 0.8421 0.8083 0.7986 0.8296 0.7975 0.7880 0.7843 0.7552 0.7468 ?= 0.1,?= 1 0.9926 1.0157 1.0165 0.9528 0.9751 0.9760 0.9498 0.9743 0.9755 0.9222 0.9469 0.9481 ?= 0.5,?= 1 0.9731 0.9895 0.9895 0.9413 0.9572 0.9572 0.9302 0.9459 0.9458 0.8970 0.9150 0.9152 ?= 0.9,?= 1 0.9012 0.8115 0.7908 0.8522 0.7722 0.7543 0.8198 0.7565 0.7402 0.7881 0.7223 0.7063 ?= 0.1,?= 3 0.9695 0.9666 0.9621 0.9391 0.9385 0.9343 0.9215 0.9226 0.9188 0.8995 0.8993 0.8953 ?= 0.5,?= 3 0.9916 0.9694 0.9617 0.9533 0.9368 0.9297 0.9340 0.9227 0.9164 0.9070 0.8965 0.8905 ?= 0.9,?= 3 0.8966 0.6667 0.6190 0.8333 0.6410 0.5952 0.8571 0.6316 0.6000 0.7975 0.6063 0.5748 ?= 0.1,?= 5 0.9660 0.9418 0.9321 0.9221 0.9014 0.8931 0.9247 0.9028 0.8941 0.8955 0.8747 0.8664 ?= 0.5,?= 5 0.9913 0.9520 0.9390 0.9533 0.9072 0.8942 0.9410 0.9081 0.8963 0.9097 0.8701 0.8580 ?= 0.9,?= 5 0.8966 0.5778 0.5306 0.8571 0.5455 0.5000 0.8214 0.5349 0.5000 0.7857 0.5238 0.4889 Table 8.I-4 of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9137 0.9997 1.0076 0.8495 0.9461 0.9612 0.8498 0.9392 0.9513 0.8287 0.9197 0.9326 ?= 0.5,?= 0.5 0.9021 0.9735 0.9739 0.8617 0.9443 0.9504 0.8390 0.9220 0.9271 0.8090 0.8950 0.9021 ?= 0.9,?= 0.5 1.0239 0.6822 0.5505 1.0005 0.6886 0.5592 0.9720 0.6675 0.5411 0.9348 0.6491 0.5276 ?= 0.1,?= 1 0.8763 0.9355 0.9231 0.8292 0.8940 0.8841 0.8287 0.8959 0.8863 0.8015 0.8664 0.8583 ?= 0.5,?= 1 0.9283 0.9531 0.9171 0.8947 0.9200 0.8878 0.8806 0.9039 0.8682 0.8519 0.8800 0.8473 ?= 0.9,?= 1 1.0284 0.5483 0.4237 1.0036 0.5384 0.4173 0.9687 0.5301 0.4097 0.9332 0.5023 0.3872 ?= 0.1,?= 3 0.8164 0.7827 0.7180 0.7918 0.7595 0.6955 0.7806 0.7501 0.6889 0.7590 0.7305 0.6706 ?= 0.5,?= 3 0.9383 0.8022 0.6979 0.8934 0.7717 0.6728 0.8767 0.7636 0.6670 0.8472 0.7448 0.6519 ?= 0.9,?= 3 1.0400 0.3210 0.2407 1.0000 0.3165 0.2404 1.0000 0.3200 0.2400 0.9341 0.3033 0.2277 ?= 0.1,?= 5 0.8048 0.7006 0.6078 0.7681 0.6762 0.5918 0.7750 0.6764 0.5888 0.7502 0.6559 0.5726 ?= 0.5,?= 5 0.9238 0.7035 0.5795 0.8844 0.6729 0.5568 0.8755 0.6731 0.5566 0.8480 0.6448 0.5321 ?= 0.9,?= 5 1.0400 0.2549 0.1926 1.0000 0.2449 0.1832 0.9583 0.2421 0.1811 0.9167 0.2366 0.1774 Table 8.I-5 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9599 0.9975 1.0010 0.9618 0.9911 0.9936 0.9600 0.9855 0.9876 0.9372 0.9661 0.9687 ?= 0.1,?= 0.9,?= 0.5 0.9729 1.0118 1.0154 0.9548 0.9863 0.9890 0.9307 0.9603 0.9629 0.9014 0.9314 0.9339 ?= 0.5,?= 0.5,?= 0.5 0.9638 0.9822 0.9833 0.9093 0.9333 0.9351 0.8896 0.9166 0.9186 0.8623 0.8886 0.8906 ?= 0.9,?= 0.1,?= 0.5 0.8905 0.8351 0.8234 0.8429 0.7986 0.7881 0.8129 0.7814 0.7721 0.7725 0.7529 0.7452 ?= 0.1,?= 0.1,?= 1 0.9896 1.0122 1.0133 0.9582 0.9806 0.9816 0.9460 0.9691 0.9701 0.9205 0.9448 0.9461 ?= 0.1,?= 0.9,?= 1 1.0038 1.0170 1.0168 0.9593 0.9787 0.9790 0.9374 0.9554 0.9557 0.9109 0.9314 0.9320 ?= 0.5,?= 0.5,?= 1 0.9807 0.9911 0.9900 0.9378 0.9517 0.9510 0.9083 0.9248 0.9246 0.8828 0.8981 0.8979 ?= 0.9,?= 0.1,?= 1 0.8947 0.7926 0.7720 0.8531 0.7740 0.7558 0.8164 0.7451 0.7283 0.7833 0.7197 0.7038 ?= 0.1,?= 0.1,?= 5 0.9732 0.9490 0.9400 0.9484 0.9245 0.9152 0.9223 0.8989 0.8900 0.9046 0.8841 0.8756 ?= 0.1,?= 0.9,?= 5 0.9920 0.9504 0.9376 0.9518 0.9100 0.8984 0.9261 0.8909 0.8798 0.9080 0.8755 0.8648 ?= 0.5,?= 0.5,?= 5 0.9881 0.9263 0.9112 0.9412 0.8922 0.8781 0.9200 0.8737 0.8606 0.8924 0.8488 0.8359 ?= 0.9,?= 0.1,?= 5 0.9310 0.5625 0.5294 0.8621 0.5435 0.5102 0.8276 0.5333 0.5000 0.7857 0.5116 0.4783 BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 3?l M.Mohammadipour, 2009, Appendix 8.I 327 Table 8.I-6 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.8966 0.9951 1.0095 0.8618 0.9532 0.9666 0.8410 0.9323 0.9458 0.8194 0.9166 0.9328 ?= 0.1,?= 0.9,?= 0.5 0.8946 0.9929 1.0052 0.8521 0.9468 0.9584 0.8340 0.9284 0.9387 0.8031 0.8964 0.9079 ?= 0.5,?= 0.5,?= 0.5 0.9370 0.9940 0.9853 0.8716 0.9462 0.9462 0.8430 0.9201 0.9190 0.8263 0.8974 0.8935 ?= 0.9,?= 0.1,?= 0.5 1.0438 0.6973 0.5664 0.9928 0.6744 0.5470 0.9687 0.6647 0.5385 0.9267 0.6488 0.5259 ?= 0.1,?= 0.1,?= 1 0.8902 0.9511 0.9374 0.8435 0.9077 0.8970 0.8267 0.8905 0.8809 0.8091 0.8737 0.8649 ?= 0.1,?= 0.9,?= 1 0.9096 0.9439 0.9177 0.8608 0.9032 0.8812 0.8485 0.8884 0.8635 0.8222 0.8629 0.8402 ?= 0.5,?= 0.5,?= 1 0.9459 0.9530 0.9039 0.8983 0.9055 0.8584 0.8810 0.8928 0.8493 0.8492 0.8602 0.8193 ?= 0.9,?= 0.1,?= 1 1.0289 0.5346 0.4123 0.9976 0.5324 0.4109 0.9667 0.5169 0.3985 0.9323 0.5015 0.3862 ?= 0.1,?= 0.1,?= 5 0.8252 0.7119 0.6190 0.8066 0.6912 0.5989 0.7825 0.6771 0.5887 0.7676 0.6695 0.5832 ?= 0.1,?= 0.9,?= 5 0.8871 0.6957 0.5817 0.8536 0.6805 0.5728 0.8240 0.6598 0.5556 0.8071 0.6483 0.5468 ?= 0.5,?= 0.5,?= 5 0.9490 0.6764 0.5492 0.8956 0.6520 0.5311 0.8804 0.6434 0.5246 0.8565 0.6216 0.5054 ?= 0.9,?= 0.1,?= 5 1.0385 0.2523 0.1901 1.0000 0.2475 0.1852 1.0000 0.2424 0.1818 0.9167 0.2292 0.1732 Table 8.I-7 of lead-time forecasts for INARMA(0,0) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.3 0.9262 0.9793 0.9846 0.8890 0.9311 0.9352 0.9549 0.9871 0.9898 0.9510 0.9776 0.9798 ?= 0.5 0.9864 1.0281 1.0313 0.9468 0.9776 0.9796 0.9496 0.9780 0.9796 0.9138 0.9451 0.9471 ?= 0.7 0.9598 1.0038 1.0066 0.9614 0.9907 0.9919 0.9425 0.9644 0.9648 0.8939 0.9233 0.9247 ?= 1 0.9710 1.0065 1.0077 0.9321 0.9571 0.9572 0.9145 0.9361 0.9356 0.8751 0.8965 0.8962 ?= 3 0.9355 0.9117 0.9022 0.8895 0.8703 0.8619 0.8704 0.8593 0.8518 0.8250 0.8159 0.8091 ?= 5 0.9362 0.8819 0.8666 0.8871 0.8368 0.8226 0.8589 0.8142 0.8004 0.8114 0.7758 0.7635 ?= 20 0.9167 0.7097 0.6667 0.8750 0.6774 0.6563 0.8696 0.6667 0.6250 0.7826 0.6000 0.5806 Table 8.I-8 of lead-time forecasts for INMA(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9733 1.0270 1.0317 0.9555 0.9873 0.9895 0.9460 0.9688 0.9699 0.9049 0.9360 0.9382 ?= 0.5,?= 0.5 0.9523 1.0027 1.0072 0.9524 0.9850 0.9874 0.9383 0.9763 0.9792 0.8847 0.9200 0.9227 ?= 0.9,?= 0.5 0.9755 1.0382 1.0438 0.9576 0.9981 1.0012 0.9222 0.9570 0.9594 0.8849 0.9195 0.9219 ?= 0.1,?= 1 0.9687 0.9911 0.9911 0.9277 0.9523 0.9526 0.9085 0.9300 0.9298 0.8799 0.9028 0.9028 ?= 0.5,?= 1 0.9863 1.0066 1.0061 0.9355 0.9576 0.9575 0.9192 0.9416 0.9414 0.8598 0.8835 0.8837 ?= 0.9,?= 1 1.0103 1.0168 1.0145 0.9382 0.9550 0.9541 0.9144 0.9405 0.9405 0.8626 0.8844 0.8842 ?= 0.1,?= 3 0.9560 0.9298 0.9205 0.8952 0.8736 0.8655 0.8626 0.8468 0.8394 0.8220 0.8114 0.8048 ?= 0.5,?= 3 0.9711 0.9415 0.9315 0.9095 0.8958 0.8877 0.8734 0.8574 0.8500 0.8389 0.8249 0.8177 ?= 0.9,?= 3 0.9943 0.9640 0.9526 0.9231 0.9063 0.8971 0.8925 0.8704 0.8609 0.8399 0.8205 0.8119 ?= 0.1,?= 5 0.9580 0.9208 0.9069 0.8882 0.8398 0.8256 0.8529 0.8147 0.8021 0.8188 0.7821 0.7696 ?= 0.5,?= 5 0.9740 0.9160 0.8988 0.9034 0.8376 0.8217 0.8770 0.8353 0.8218 0.8360 0.7921 0.7788 ?= 0.9,?= 5 0.9902 0.9220 0.9034 0.9071 0.8420 0.8254 0.8766 0.8191 0.8032 0.8424 0.7897 0.7751 BenchmarkINARMA MASEMASE / )( 3?l BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l M.Mohammadipour, 2009, Appendix 8.I 328 Table 8.I-9 of lead-time forecasts with smoothing parameter 0.2 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.9080 0.9615 0.9663 0.9518 0.9839 0.9863 0.9521 0.9779 0.9794 0.9064 0.9400 0.9424 ?= 0.5,?= 0.5 0.9824 1.0279 1.0320 0.9527 0.9852 0.9877 0.9140 0.9488 0.9514 0.8656 0.9005 0.9032 ?= 0.9,?= 0.5 1.0092 0.9926 0.9827 0.9498 0.9300 0.9211 0.9263 0.9104 0.9027 0.8765 0.8602 0.8525 ?= 0.1,?= 1 0.9865 1.0081 1.0078 0.9450 0.9617 0.9612 0.9151 0.9374 0.9374 0.8687 0.8929 0.8933 ?= 0.5,?= 1 1.0050 1.0159 1.0145 0.9551 0.9701 0.9692 0.9125 0.9373 0.9378 0.8690 0.8939 0.8942 ?= 0.9,?= 1 1.0143 0.9115 0.8898 0.9557 0.8725 0.8532 0.9240 0.8675 0.8519 0.8679 0.8054 0.7894 ?= 0.1,?= 3 0.9546 0.9386 0.9306 0.9051 0.8913 0.8837 0.8688 0.8543 0.8468 0.8307 0.8222 0.8158 ?= 0.5,?= 3 0.9990 0.9675 0.9572 0.9385 0.9149 0.9059 0.9088 0.8915 0.8833 0.8642 0.8515 0.8437 ?= 0.9,?= 3 1.0204 0.7576 0.7246 0.9388 0.7419 0.7077 0.9375 0.7258 0.6923 0.8723 0.6949 0.6613 ?= 0.1,?= 5 0.9512 0.9068 0.8926 0.8869 0.8411 0.8280 0.8570 0.8155 0.8028 0.8227 0.7847 0.7726 ?= 0.5,?= 5 1.0021 0.9556 0.9397 0.9286 0.8686 0.8528 0.9026 0.8527 0.8380 0.8573 0.8159 0.8026 ?= 0.9,?= 5 1.0000 0.7143 0.6250 1.0000 0.7143 0.6250 0.8000 0.5714 0.5000 0.8000 0.5714 0.5714 Table 8.I-10 of lead-time forecasts with smoothing parameter 0.5 for INAR(1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.5 0.8561 0.9604 0.9661 0.8196 0.9151 0.9187 0.7849 0.8858 0.8931 0.7399 0.8464 0.8567 ?= 0.5,?= 0.5 0.9043 1.0091 1.0141 0.8535 0.9527 0.9551 0.8062 0.9129 0.9190 0.7588 0.8601 0.8668 ?= 0.9,?= 0.5 1.0597 0.8030 0.6621 1.0240 0.7841 0.6514 1.0063 0.7790 0.6469 0.9540 0.7399 0.6155 ?= 0.1,?= 1 0.8545 0.9039 0.8707 0.7854 0.8350 0.8092 0.7494 0.8087 0.7858 0.7065 0.7673 0.7476 ?= 0.5,?= 1 0.8989 0.9150 0.8729 0.8459 0.8734 0.8343 0.8072 0.8468 0.8139 0.7685 0.8075 0.7742 ?= 0.9,?= 1 1.0649 0.6216 0.4873 1.0164 0.6048 0.4736 1.0082 0.6263 0.4909 0.9450 0.5712 0.4472 ?= 0.1,?= 3 0.7493 0.6930 0.6140 0.7068 0.6516 0.5799 0.6679 0.6167 0.5491 0.6462 0.6053 0.5406 ?= 0.5,?= 3 0.8756 0.7363 0.6295 0.8082 0.6997 0.6050 0.7869 0.6843 0.5917 0.7447 0.6508 0.5625 ?= 0.9,?= 3 1.0417 0.3817 0.2874 1.0222 0.3770 0.2822 1.0000 0.3719 0.2795 0.9535 0.3596 0.2715 ?= 0.1,?= 5 0.7364 0.6034 0.5083 0.6842 0.5666 0.4794 0.6585 0.5442 0.4584 0.6361 0.5287 0.4476 ?= 0.5,?= 5 0.8716 0.6526 0.5337 0.7972 0.5955 0.4873 0.7758 0.5884 0.4812 0.7404 0.5633 0.4614 ?= 0.9,?= 5 1.0000 0.2941 0.2273 1.0000 0.3125 0.2381 1.0000 0.2667 0.1905 1.0000 0.2667 0.2105 Table 8.I-11 of lead-time forecasts with smoothing parameter 0.2 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.9190 0.9654 0.9696 0.9551 0.9896 0.9920 0.9387 0.9724 0.9748 0.8993 0.9321 0.9346 ?= 0.1,?= 0.9,?= 0.5 1.0104 1.0541 1.0576 0.9574 0.9895 0.9918 0.9131 0.9504 0.9532 0.8746 0.9094 0.9120 ?= 0.5,?= 0.5,?= 0.5 1.0190 1.0438 1.0452 0.9688 0.9964 0.9980 0.9232 0.9541 0.9562 0.8675 0.9024 0.9051 ?= 0.9,?= 0.1,?= 0.5 1.0060 0.9521 0.9398 0.9600 0.9113 0.9004 0.9176 0.8894 0.8806 0.8607 0.8460 0.8388 ?= 0.1,?= 0.1,?= 1 0.9892 1.0070 1.0065 0.8959 0.8533 0.8398 0.9082 0.9370 0.9377 0.8690 0.8946 0.8951 ?= 0.1,?= 0.9,?= 1 1.0118 1.0259 1.0245 0.9382 0.9564 0.9557 0.9108 0.9328 0.9327 0.8610 0.8808 0.8805 ?= 0.5,?= 0.5,?= 1 1.0204 1.0394 1.0385 0.9611 0.9754 0.9741 0.9162 0.9332 0.9325 0.8625 0.8835 0.8833 ?= 0.9,?= 0.1,?= 1 1.0111 0.9258 0.9037 0.9531 0.8750 0.8556 0.9201 0.8685 0.8520 0.8585 0.8062 0.7913 ?= 0.1,?= 0.1,?= 5 0.9550 0.9088 0.8948 0.8946 0.8574 0.8447 0.8537 0.8125 0.8000 0.8299 0.7949 0.7830 ?= 0.1,?= 0.9,?= 5 0.9852 0.9053 0.8866 0.9208 0.8598 0.8429 0.8834 0.8362 0.8213 0.8365 0.7916 0.7778 ?= 0.5,?= 0.5,?= 5 1.0278 0.9487 0.9250 0.9444 0.8718 0.8718 0.9167 0.8462 0.8250 0.8571 0.8108 0.7895 ?= 0.9,?= 0.1,?= 5 1.0000 0.6250 0.6250 1.0000 0.7143 0.6250 0.8000 0.5714 0.5000 0.8000 0.5714 0.5714 BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l BenchmarkINARMA MASEMASE / )( 6?l M.Mohammadipour, 2009, Appendix 8.I 329 Table 8.I-12 of lead-time forecasts with smoothing parameter 0.5 for INARMA(1,1) series Parameters ?=?? ?=?? ?=?? ?=?? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? Croston ?=?.? SBA ?=?.? SBJ ?=?.? ?= 0.1,?= 0.1,?= 0.5 0.8436 0.9460 0.9554 0.8118 0.9176 0.9250 0.7733 0.8817 0.8922 0.7361 0.8400 0.8521 ?= 0.1,?= 0.9,?= 0.5 0.9077 1.0043 1.0022 0.8329 0.9281 0.9292 0.7716 0.8820 0.8923 0.7343 0.8368 0.8437 ?= 0.5,?= 0.5,?= 0.5 0.9502 1.0183 1.0071 0.8870 0.9706 0.9628 0.8296 0.9157 0.9126 0.7871 0.8756 0.8742 ?= 0.9,?= 0.1,?= 0.5 1.0448 0.7635 0.6331 1.0426 0.7649 0.6361 0.9998 0.7685 0.6390 0.9469 0.7337 0.6099 ?= 0.1,?= 0.1,?= 1 0.8417 0.8799 0.8498 0.6944 0.5712 0.4790 0.7451 0.8140 0.7957 0.7078 0.7706 0.7513 ?= 0.1,?= 0.9,?= 1 0.8727 0.9003 0.8607 0.7934 0.8331 0.8001 0.7713 0.8107 0.7805 0.7290 0.7629 0.7318 ?= 0.5,?= 0.5,?= 1 0.9169 0.9273 0.8766 0.8656 0.8799 0.8314 0.8179 0.8400 0.7997 0.7738 0.7972 0.7581 ?= 0.9,?= 0.1,?= 1 1.0504 0.6051 0.4690 1.0216 0.6003 0.4673 0.9949 0.6140 0.4789 0.9411 0.5804 0.4538 ?= 0.1,?= 0.1,?= 5 0.7387 0.6121 0.5152 0.6994 0.5795 0.4883 0.6619 0.5486 0.4636 0.6480 0.5389 0.4545 ?= 0.1,?= 0.9,?= 5 0.8082 0.5982 0.4868 0.7500 0.5670 0.4637 0.7260 0.5548 0.4544 0.6815 0.5287 0.4350 ?= 0.5,?= 0.5,?= 5 0.9024 0.6379 0.5139 0.8293 0.5965 0.4789 0.8049 0.5789 0.4714 0.7500 0.5556 0.4478 ?= 0.9,?= 0.1,?= 5 1.0000 0.2941 0.2174 1.0000 0.3125 0.2273 1.0000 0.2500 0.1905 1.0000 0.2667 0.2000 BenchmarkINARMA MASEMASE / )( 6?l M.Mohammadipour, 2009, Appendix 9.A 330 Appendix 9.A INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) Series of 16,000 Series In this appendix, the identified INARMA series of 16,000 data set are separated and forecasted with the known INARMA models. Investigating the estimated parameter of the INARMA(0,0) process (?), we found that in general ? is close to 0.1 (the average is 0.1953 and 69.26 percent are between 0 and 0.15). The estimated autoregressive parameter of the INAR(1) process, ?, is close to 0.2 (the average is 0.2460 and 52.94 percent are between 0.1 and 0.3) and the estimated innovation parameter, ?, is around 0.5 (the average is 0.3562 and 97.06 percent are between 0 and 1). Looking at the estimated parameter of the INMA(1) process (?,?) reveals that in general, ? is close to zero (the average is 0.0898 and 46.29 percent are between 0 and 0.1) and ? is around 0.3 (the average is 0.3782 and 55.56 percent are between 0.2 and 0.4). In general, the estimated autoregressive parameter of an INARMA(1,1) process is in the range 0.1